1130
J. Phys. Chem. B 1997, 101, 1130-1137
Computational Study of Iron Hexacyanide in Silver Halide R. C. Baetzold Imaging Research and AdVanced DeVelopment, Eastman Kodak Company, Rochester, New York 14650-2021 ReceiVed: September 16, 1996; In Final Form: NoVember 25, 1996X
Classical and quantum mechanical atomistic calculations are presented for the structure and energy levels of iron hexacyanide complexes in silver halide crystals. The classical calculations employ a shell-model interatomic potential for the silver halide host and a force field developed from the vibrational properties of Fe(CN)64- and Fe(CN)63-. In addition to the normal solid state treatments it was necessary to include polarizability of the CN- ligands by means of the shell model and a bonding Ag+-N interatomic potential derived quantum mechanically in order to predict single and double silver ion vacancy orientations near the hexacyanide that could be rationalized with experimental interpretations. In AgCl, single vacancies were predicted to occupy one of the six equivalent (200) positions near Fe(CN)64-, and near Fe(CN)63- the divacancy configurations (200) (2h00); (2h00) (020); and (1h1h0) (200) were predicted to be favored. The electronic properties of the dopant-vacancy complexes in silver halide were calculated with embedded Hartree-Fock and local density methods. It is shown that Fe(CN)64- is not a deep electron trap since its calculated electron affinity is less than that calculated for the AgCl host. Thus, this complex can only function as a shallow ionized donor center. There was a perturbation of the charge distribution calculated for the complex caused by the presence of associated cation vacancies. These complexes have an ionization potential calculated to be less than that of AgCl, which permits them to trap holes.
1. Introduction Recent reports1,2 have indicated the usage of the ioniccovalent dopant iron hexacyanide in photographic studies in AgCl. EPR has proven1,2 to be a valuable experimental tool for studying this dopant-host system. The experiments have shown that the dopant substitutes intact for AgCl65- and is well dispersed in the crystal lattice. The Fe(CN)64- charge state of the dopant was shown1,2 to possess amphoteric properties. It functions as a shallow electron trap, or ionized donor center, when free, but becomes a hole trap when compensated by association with a silver ion vacancy. Cation vacancies are present in silver halide through the Frenkel equilibrium, which is dominant in these materials. A second vacancy may also become associated with the complex following the hole trapping step. The equilibria controlling these association processes are functions of temperature and the vacancy binding energies. An important question involves the structure of the free and vacancy associated Fe(CN)64- or Fe(CN)63- dopant in AgCl. The crystal lattice must undergo a degree of distortion in order to accommodate this bulky ionic-covalent dopant. Then, in order for vacancies to become associated with the complex, additional lattice distortion must take place. There have been several studies of this and related dopants in alkali halide3 crystals. EPR has clearly shown the association of cation vacancies at the next and next-nearest-neighbor sites to the dopant. These studies are significant in that single crystals can be used, which greatly aids the ability to make unequivocal structural assignments. In one recent study,4 EPR and ENDOR have been used to determine the structure of divacancy-Fe(CN)63- complexes in single-crystal NaCl. Unfortunately, doping of the complex into single crystals of silver halide has not been possible, so that similar detailed structural information has not been experimentally accessible. The positions of the energy levels of Fe(CN)64- in AgCl have not been directly measured, but a crystal-field model has been X
Abstract published in AdVance ACS Abstracts, January 15, 1997.
S1089-5647(96)02844-1 CCC: $14.00
used1,2 successfully to understand its electron and hole trapping behavior. The d6 complex should have a filled t2g manifold separated by 10 Dq5 of 4.2 eV from the unfilled eg levels. An ability to trap holes implies that the t2g levels lie in the AgCl band gap region, while the 10 Dq value, which exceeds the band gap of 3.2 eV, implies that the eg levels lie above the conduction band edge. These qualitative considerations do not account for possible shifts in levels due to vacancies but have provided a useful working approximation to the system. The purpose of this report is to establish how computer simulation methodologies can be applied to the iron hexacyanide dopant in silver halide. A variety of computational as well as structural and energetic issues surrounding the behavior of this dopant must be addressed. We apply the Hartree-Fock and local density quantum mechanical methods to determine carrier trap depths and structure, but the cluster must be coupled to the lattice crystal ions. Correlation energy effects are computed with the perturbation methods following the Hartree-Fock calculation. An interatomic potential for iron hexacyanide in silver halide is developed and applied to determine the structure of vacancies near the dopant. Comparison to available experimental data is used to test different levels of approximation in these calculations. 2. Method Atomistic Simulations. The structure and binding energy of cation vacancies to iron hexacyanide complexes in silver halide have been computed by classical atomistic simulations. We employ the Mott-Littleton methodology which is incorporated in the CASCADE6a or GULP6b computer codes. This procedure requires the availability of accurate interatomic potentials capable of describing the dopant, the ionic crystal, and the interactions between the two. A force field has been previously derived7 for iron hexacyanide based upon its vibrational spectra. The original force field was composed of harmonic bond stretching and bending terms. The bond bending terms were used in their original form © 1997 American Chemical Society
Iron Hexacyanide in Silver Halide 1
/2K(θ - θ0)2
J. Phys. Chem. B, Vol. 101, No. 7, 1997 1131
(1)
where K is a force constant and θ, θ0 represent the actual and reference angles between bonds. The bond stretching function was fitted to a Morse function
V(r) ) D(1 - e-β(ri-ri0))2
(2)
where ri and ri0 are actual and reference bond lengths, D is the bond dissociation energy, and β is derived from the harmonic force constants. These parameters are reported in Tables 8-10. The interatomic potentials for AgCl and AgBr have been empirically derived8,9 based upon properties of the crystal. Coulomb interactions are treated for ions of integer charge that are represented by a core and shell of different charge that are coupled harmonically. The shells interact through short-range potentials that represent the overlap of electron clouds from adjacent ions. These potentials contain two-body terms
Ae-r/F - C/r6
(3)
where r is the distance between shells and A, F, and C are constants. Three-body terms can also be included. Previous work8 has defined bond harmonic functions in the form of eq 1 and triple-dipole functional forms. These potentials have been used with some constants adjusted from the previous work.8 This form of potential has provided8 excellent agreement of elastic, dielectric, and defect properties of silver halide with experiment. The interactions between dopant and silver halide include Coulombic and short-range terms. The Coulombic terms arise from the charge distribution in the crystal and in the dopant that was determined from the Mulliken charges in a HartreeFock calculation. This assignment of ion charges is somewhat arbitrary, as is known for Mulliken analyses, but consistent with other analyses based upon multipole expansions of the electron density.10 The short-range interaction terms are represented as a Born-Mayer potential
A e-r/F
(4)
The constants in this potential were computed using the electron-gas approximation.11 All of the constants of these potentials are given in Tables 8-10. This potential has provided a good representation for the dopant-silver halide interactions with the exception of Ag+-N interactions involving cyanide. In this case we computed the interaction of CN- with a silver ion using AgCl43- embedded within a hemispherical array of point ions. The Hartree-Fock method involving MP2 electron correlation treatment was employed. Two orientations were considered involving the atoms C-N-Ag at 180° or 90° in a σ or π type of arrangement. The resulting interaction energies showed potential energy minima vs the N-Ag distance and were fitted to eq 3 giving the parameters in Tables 8-10. Details of our basis functions and point ion arrays are described under the quantum mechanical section. An atomistic classical procedure based upon the MottLittleton approximation12,13 is used to calculate the interaction of dopant and cation vacancies. The method has been discussed in detail14,15 so comments here will be brief. The dopant molecule is enclosed by a spherical region, variable in size, but having a radius of seven nearest-neighbor distance. The ions in this region are described by the shell model.16 Each ion is represented by a massless shell charge harmonically coupled to a core charge having the appropriate atomic mass. The core
and shell charges add to give the formal ion charge and along with the harmonic constant determine the perfect crystal dielectric constants. The core and shell can separate slightly in order to simulate lattice polarization. For distances greater than seven nearest neighbors, continuum methods are used to represent the crystal interaction with the dopant. An equilibrium structure for the dopant within the crystal is achieved corresponding to zero force on each ion. The initial treatments of cyanide ligands in the complex were as point charges. This treatment leads to the most favorable arrangements of cation vacancies in a highly symmetric nearestneighbor positions near the dopant. Further calculations have employed the shell model on N and C atoms, which provides a polarizability to the ligands. The force constants are varied to give equal polarizabilities on C and N as determined by the squared shell charge divided by the force constant. Finally, we introduced the bonding type of Ag+-N interaction potential for comparison to results obtained with the Born-Mayer form of potential. Employing this procedure leads to favored dopant vacancy configurations that have low symmetry and which can be rationalized more readily with experimental assignments.1,2 Quantum Mechanical Simulations. The quantum mechanical calculations employ the restricted closed or open-shell Hartree-Fock procedures. We employ double-ζ plus polarization basis functions for C, N, and Cl. The Fe basis functions are (5, 2, 1, 1, 1, 1/5, 1, 1, 2/4, 1, 1) from ref 17, and the Ag basis is a valence double-ζ plus 2p + d polarization basis with a model potential.18 The computer code CADPAC19 is used for this purpose. In this calculation the iron hexacyanide molecule is embedded within a sphere of 1206 point ions having unit positive or negative charges and positioned according to the AgCl lattice. The Born-Mayer function acts between the quantum ions and the nearest point ions. The parameters are the same as those in the atomistic calculations described above. The calculation proceeds by a geometry optimization using appropriate Oh, D4h, or C4 symmetry specification depending upon the appropriate electronic state or vacancy distribution around the iron hexacyanide. The total energy includes electronic and short-range terms and is minimized by the analytical gradient method in CADPAC.19 The Hartree-Fock procedure for closed shells (SCF) and open shells (ROHF) is employed in the geometry optimization phases of the calculation. We have also examined possible effects of electron correlation by performing a Moller-Plesset second-order perturbation theory (MP2) calculation on various charge states with ions at their optimized positions. The local density method using the BLYP functional in CADPAC has also been used in the same fashion. Lattice polarization effects are determined in a second calculation. Here the optimized iron hexacyanide geometry and corresponding charge distribution is kept fixed within an atomistic simulation calculation. The change in energy accompanying the polarization and displacement of lattice ions is then calculated. Double counting of the interaction terms between lattice and dopant is avoided by subtracting the energy term due to Born-Mayer terms as determined at the optimized geometry. The polarization term is added to the quantum mechanical energy to get the total energy of the system. Ionization potentials are determined by subtracting the total energy of the appropriate fully relaxed charge states. The positions of energy levels relative to the AgCl valence band edge is calculated from the ionization energy of a AgCl65- unit embedded in point charges and having corrections made for lattice polarization like those used for iron hexacyanide. This
1132 J. Phys. Chem. B, Vol. 101, No. 7, 1997
Baetzold
Figure 1. A sketch that shows the ion positions in one plane for Fe(CN)64- substituted in AgCl. Note that nearest neighbor (110), (11h0), (1h10), and (1h1h0) sites and next-nearest-neighbor (200) sites are identified.
overall procedure for treatment of lattice polarization has been employed previously20 for metal-halide systems. An area for improvement of the interaction of the dopant complex with the lattice involves the expression of dopant complex charges in a more precise representation including dipole and quadrapole moments. Such a method has been developed and applied to several systems for treating this problem and is described21 where several of the relevant references may be found. In future work it would be important to extend the range of application of this methodology to these dopant complexes. 3. Results Fe(CN)6x- Vacancy Complexes in NaCl. We begin with studies of the hexacyanide complexes in the NaCl crystal, because of the availability of experimental structural information in this system. An interatomic potential for NaCl is taken from the literature22 and used with parameters of the complex described before. There is strong association of a cation vacancy with the Fe(CN)63- cluster, which is substituted for NaCl65-. These vacancies are present through the normal Schottky defect mechanism predominant for point defect creation in NaCl. We compute a binding energy of 1.37 eV for cation vacancies at any of the 12 equivalent nearest-neighbor positions (110) to the complex and 1.04 eV for any of the six equivalent nextnearest-neighbor positions (200). These positions are sketched for a rock salt structure shown in Figure 1. In accord with Coulomb considerations in a low dielectric crystal, the geometry having the shortest Fe2+-vacancy distance is favored. The Fe(CN)63- complex in NaCl is compensated by two associated cation vacancies. There are nine possible distinct arrangements of nearest or next-nearest cation vacancies that are sketched in Figure 2 where we follow the labeling scheme of ref 4. The binding energy of the two vacancies relative to the isolated species is computed and presented in Table 1. The configuration with the strongest binding is j ) 4 with Na+ vacancies at (110) and (1h1h0). The most populated center observed experimentally (I)4 has been assigned to this configuration. Two other centers (II, III) were observed experimentally4 and were assigned to j ) 3 and 6. We compute j ) 3 to be the second most stable orientation of vacancies and this agrees well with the experiment. The experimental assignment of j ) 6 is not in agreement with our calculated relative binding energies. The data in Table 1 would predict that j ) 1, 2, or 5 would be more stable than j ) 6. This is a small discrepancy that is not understood. The calculations may be in error, or the
Figure 2. A sketch that shows the nine possible divacancy configurations considered when Fe(CN)63- is substituted in a crystal with the NaCl structure.
TABLE 1: Binding Energy of Divacancy-Fe(CN)63Complexes in NaCl configuration, j
vacancy orientation
BE (eV)
1 2 3 4 5 6 7 8 9
01h1 01h1 1h01h 101h 1h1h0 101 110 1h1h0 110 200 011 200 110 200 200 2h00 2h00 020
2.141 2.182 2.419 2.480 2.110 2.042 1.840 1.730 1.690
experiment4
II, III I II, III
experimental assignment may be in error. The assignment was based on the assumption that the complex with vacancies remains octahedral, while deviations from this symmetry were found in the calculations. We note that ENDOR was not applied to the assignment in question. Another possibility is that the complexes may not be fully equilibrated. Overall, the agreement between experimentally observed species and calculated relative binding energies is good. We can consider the distortions induced in the Fe(CN)63complex through insertion in the NaCl lattice. The complex remains octahedral and is compressed in the absence of vacancies and takes on bond lengths of Fe-C ) 2.00 Å and C-N ) 1.15 Å. In the presence of vacancies the complex can become distorted as shown in Figure 3 for j ) 4. The cyanide ligands are repelled by the vacancies, and they are rotated by θ ) 7.4° away from their normal position. Experimental4 ENDOR data have given a distortion similar to the calculated one with a rotation of 6-8°, in good agreement with calculation. This level of agreement suggests that we can treat the interactions of the bulky ionic-covalent Fe(CN)63- within an ionic lattice well. Geometry of Hexacyanide Complexes in AgCl. The hexacyanide complex Fe(CN)64- occupies more volume than the AgCl65- unit which it replaces in AgCl. The calculated Fe2+-C bond length is 2.08 Å, and C-N bond length is 1.15 Å when the system is fully relaxed. The sum of these distances (3.23 Å) exceeds the normal Ag-Cl distance of 2.753 Å. This complex, in the absence of association with vacancies, remains octahedral in shape. Lattice ions are distorted from their perfect
Iron Hexacyanide in Silver Halide
J. Phys. Chem. B, Vol. 101, No. 7, 1997 1133 TABLE 3: Binding Energy (eV) of Vacancies to Fe(CN)64in Silver Halide Using Bonding Ag-N Potential Function BE (eV) material
two-body
AgCl
triple-dipole
AgBr
two-body
AgBr Figure 3. A sketch that shows the distortions of cyanide ligands within a plane when the Fe(CN)63- complex is relaxed in the NaCl crystal in the presence of (110) and (1h1h0) cation vacancies.
Figure 4. A sketch showing the relaxation of lattice ions around a cyanide ligand when Fe(CN)64- is substituted in AgCl.
TABLE 2: Calculated Binding Energy (eV) of Vacancies to Fe(CN)64- in Silver Chloride BE (eV) potential two-body
bond-harmonic
triple-dipole
a
3
CN polarizability (Å )
(110)
(200)
10.63 2.13 1.06 0.00 10.63 2.13 1.06 0.00 10.63 2.13 1.06 0.00
0.60 0.75 0.67 0.54 0.68 0.72 0.54 0.48 0.86 0.77 0.62 0.40
0.53 0.47 0.31 0.38 0.56 0.45 0.26 0.37 0.77 0.58 0.35 0.44
-
a Polarizability calculated from shell charge squared divided by force constant.
positions. A sketch of the displacements for the closest ions is shown in Figure 4. Silver ions at (200), toward which the cyanide ligands point, relax 0.22 Å away from the complex. Along the Cartesian axes the ions move away from the complex, but off-axis there is a small reverse displacement. The largest Cl- ion displacement is 0.21 Å toward the complex from lattice ions originally at (111). We have examined various approximations in our calculation in order to determine which give reliable predictions of the favored vacancy configuration near Fe(CN)64- in AgCl since there is no precidence for treating this system. Experimentally,1,2 the (200) vacancy complex is favored over the (110) vacancy. We examined both the effects of CN- polarizability and Ag+-N bonding interactions on the predicted orientation. In Table 2 we report values of the vacancy binding energy calculated for various CN- polarizability values using the electron gas potential of eq 4 for Ag+-N interactions. The force constant is varied in order to simulate different C and N polarizabilities which are taken equal. Generally, the binding energy of the vacancy increases as the polarizability of CNincreases, but the (110) geometry is favored over the (200) geometry with only one exception. The range of polarizability
potential
AgCl
bond-harmonic
CN polarizability Å -
10.63 2.13 1.06 0.00 10.63 2.13 1.06 0.00 10.63 2.13 1.06 0.00 10.63 2.13 1.06 0.00
3
(110)
(200)
0.42 0.35 0.22 0.12 0.44 0.34 0.15 0.06
0.53 0.35 0.17 0.29 0.51 0.51 0.25 0.34
0.35 0.23 0.19 0.45 0.42 0.33 0.24
0.24 0.10 0.09 0.41 0.28 0.20 0.29
values studied varies from values appropriate to halides in an ionic crystal to zero. The values calculated for the different AgCl potentials agree with one another in relative binding energies. We now include the bonding Ag+-N interaction potential in our classical calculations in order to evaluate which vacancy configuration is preferred near Fe(CN)64- in silver halide. The results of these calculations using different CN- polarizabilities are shown in Table 3. The (200) vacancy configuration is preferred over (110) for most of the entries with the different AgCl potentials. Results from the triple-dipole potential strongly favor the (200) configuration. For AgBr, we report results for two different potentials and here the (110) configuration is generally favored over the (200) configuration. This represents a clear difference from AgCl but is in accord with experiments.1,2 We conclude that the reason for observation of the (200) vacancy near Fe(CN)64- in AgCl is involved with a covalent bonding between the cyanide ligands and nearest silver ions with some dependence on the silver halide type and that this feature must be included in interatomic potentials treating this problem. The binding of divacancies to Fe(CN)63- in AgCl and AgBr was treated at the different levels of calculation described before. We first employed the repulsive Ag+-N potential with twobody and bond-harmonic potentials for AgCl. Representative results are plotted in Figure 5. A range of CN- polarizabilities was considered, and the total binding energy increased with polarizability as observed for single vacancies with Fe(CN)64-. The symmetric (110) (1h1h0) configuration is preferred if the CNis not polarized, but (1h1h0) (101) and (1h1h0) (200) become favored with greater polarizability. Experimental assignments1,2 indicate (200) (2h00) as a minority, but definite species present in AgCl along with three lower symmetry structures. This result seems to indicate that this level of approximation is inadequate for treating divacancies near Fe(CN)63- in AgCl because this level of calculation predicts little relative stability for this configuration. We turn to the bonding Ag+-N interatomic potential function for treatment of divacancies near Fe(CN)63- in silver halide. Results are shown in Table 4 for the various potentials. The CN- polarizability is kept at 2.13 Å3 for these comparisons. For AgCl, each of the three interatomic potentials predicts that the (2h00) (020) configuration is favored. The other two configurations that have the largest binding energies are (200) (2h00) and (1h1h0) (200). As mentioned earlier, the (200) (2h00) divacancy configuration has been identified experimentally.1,2 This level of calculation agrees with experiment on the stability
1134 J. Phys. Chem. B, Vol. 101, No. 7, 1997
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3-
Figure 5. Binding energy of two silver ion vacancies to Fe(CN)6 in AgCl computed with the two-body and bond-harmonic potential for AgCl and repulsive Ag+-N potential of eq 4 in the text. We consider various CN- polarizability values for the nine configurations described in the text.
Figure 7. A sketch showing the relaxed positions of C atoms (in cyanide ligands) for Fe(CN)63- substituted in AgCl in the presence of (1h1h0) and (101) cation vacancies. The plane of the relaxed ion is denoted by each ligand distorted from a Cartesian axis.
Figure 8. Sketches showing the displacement of the Fe ion in Fe(CN)63- in AgCl with four different divacancy orientations computed with the triple-dipole potential.
Figure 6. Binding energy of two silver ion vacancies to Fe(CN)63- in AgCl computed with the triple-dipole and two-body potentials for AgCl and bonding Ag+-N potential of eq 3 in the text. We consider various CN- polarizability values for the nine configurations described in the text.
TABLE 4: Binding Energy of Silver Ion Vacancies with Fe(CN)63- in Silver Halide for CN- Polarizability of 2.13 Å3 Using Bonding Ag-N Potential Functions config
vacancies
AgCla
AgClb
AgClc
AgBrd
AgBre
1 2 3 4 5 6 7 8 9
1h01 01h1 1h01h 101h 1h1h0 101 1h1h0 110 1h1h0 200 011 200 110 200 200 2h00 2h00 020
1.34 NC 1.52 1.31 1.62 1.52 1.41 1.53 1.65
0.94 1.01 1.02 0.95 1.25 1.18 1.04 1.16 1.27
1.37 1.45 1.66 1.41 1.83 1.75 1.55 1.74 1.95
1.04 NCf 1.22 1.17 1.28 1.18 1.06 1.16 1.23
1.12 1.24 1.27 1.27 1.31 1.25 1.18 1.22 1.27
a Two-body AgCl potential, ref 9. b Bond-harmonic AgCl potential, ref 8. c Triple-dipole AgCl potential, ref 8. d Two-body AgBr potential, ref 9. e Bond-harmonic potential, ref 8. f NC ) not converged.
of the (200) (2h00) configuration in AgCl. For AgBr, these three divacancy configurations are favorable as well as some of the others including (110) (1h1h0) and (1h1h0) (101). It appears that some additional orientations are possible for AgBr. We have also examined the effects of CN- polarizability on these results. Figure 6 shows this effect as computed for AgCl with the tripledipole and two-body potentials. There are small effects in the
relative stability of the different complexes, but the three species identified above remain predominant. We believe that these results can be used to assign the species found in experiment. There is a complex pattern of dopant distortions around the lower symmetry divacancy complexes. We show an example of this in Figure 7 for the (1h1h0), (101) divacancies near Fe(CN)63- in AgCl. The nearest C ions are relaxed considerably so that there is a deviation of the C-Fe-C bond angle from 180°. In addition, there is a displacement of the iron ion from the silver ion site of the AgCl65- substitutional unit. This behavior is shown for the favored orientations calculated for AgCl in Figure 8. Displacements of up to about 0.22 Å are observed for the favored (2h00) (020) geometry. For the (200) (2h00) geometry, the Fe ion remains at the origin. Geometry of Iron Hexacyanide in AgBr. The iron hexacyanide substituted for AgBr65- in AgBr gives a similar pattern of lattice ion distortion in AgBr and AgCl. The relaxed Fe-C distance of 2.12 Å and C-N of 1.16 Å exceeds the AgBr distance by about 0.4 Å. Unlike AgCl, the (110) vacancy orientation near Fe(CN)64- is preferred in AgBr for most values of the CN- polarizability. When we include the bonding Ag+-N potential, the absolute values of the binding energy of a vacancy to Fe(CN)64- are calculated to be comparable in AgCl and AgBr using the two-body potentials. Comparison of the total energy of the divacancy complexes near Fe(CN)63- in Table 4 gives weaker binding in AgBr than AgCl when we employ the two-body potentials. This is consistent with a larger dielectic constant in AgBr. Shallow Electron Trapping. Shallow electron trapping by Coulombic centers is well documented in the silver halides for
Iron Hexacyanide in Silver Halide
J. Phys. Chem. B, Vol. 101, No. 7, 1997 1135
interstitial silver ions,23 impurities,24 and the iron hexacyanides.1,2 The effective mass approximation25 has proven to give a good description of the trap depths and transition energies of the electron. In this approximation the binding energy is given by
En )
-e4m*z2 22p2n2
(5)
where e is the electron charge, z is the nuclear charge, m* is the electron effective mass, is the dielectric constant, p is Planck’s constant, and n is the quantum number. In this approximation the Bohr radius of the electron is
rn )
n2p2 e2m*z
(6)
These formulas give 0.065 eV as the ground state binding energy and 12 Å as the first Bohr radius26 in AgCl at low temperature. The substitutional iron hexacyanide dopant in AgCl is unlike the point charge defects mentioned above for shallow electron traps. In iron hexacyanide the charge is delocalized over the iron and cyanide ligands as may be seen in Tables 8-10. Also, there is a distortion of the silver halide lattice by the introduction of this dopant. Thus, it is of interest to compute the trap depths for this dopant by alternative means in order to compare to the effective mass approximation. In order to facilitate the comparison, we substitute eq 6 into eq 5, neglecting the quantized nature of these solutions. For the ground state
E ) -e2z/2r
(7)
We plot the binding energy for the z ) 1 ground state computed from eq 7 in Figure 9 as a reference for comparison to the calculated binding energies from various dopants. We have computed the binding energy of an electron to the iron hexacyanide dopants using the atomistic procedure. We simulate the conduction electron as Ag0 on a cation lattice site placed at varying distances from the dopant. This type of polaron model has been used previously27 to compute the band gap in insulators, and we will comment on this approximation later. This procedure gives a binding energy that is just the difference in energy of the isolated and
BE ) E(complex) + E(Ag0) - E(complex + Ag0) (8) interacting components of the system. The binding energy is plotted vs distance between the electron and Fe ion in the complex in Figure 9. The uncompensated Fe(CN)64- complex follows the effective mass approximation well, indicating its shallow electron trapping character. The uncompensated Fe(CN)63- complex has a deeper binding because of its greater attraction for the electron and becomes a somewhat deeper electron trap, but one which functions over long distances. Here we are not permitting deep trapping through the hole in the t2g levels. The Fe(CN)63- plus one cation vacancy complex presents a somewhat shallow electron trap. The vacancy is shielding the positive charge of the complex very effectively. Finally, the [Fe(CN)64- plus one cation vacancy] and [Fe(CN)63plus two cation vacancies] complexes are not electron traps over the distance scale where the shallow traps function. Here the vacancies completely shield the positive charge of the complex. It is interesting to point out that the theory of deep electron trapping is thought to first proceed through electron trapping into the shallow trapped state. In this regard, even though the
Figure 9. Binding energy of a shallow trapped electron to various Fe(CN)6 dopants substituted in AgCl is shown as a function of distance. The behavior of the effective mass approximation is also plotted. Attraction is indicated for positive binding energies.
TABLE 5: Shallow Electron Trapping Binding Energy (eV) for Ag0 vs Delocalized Electrons for Fe(CN)64-/AgCl distance (Å)
delocalized
Ag0
7.8 9.5 11.0
0.19 0.14 0.14
0.15 0.14 0.14
Fe is present in the +3 oxidation state in [Fe(CN)63- plus two vacancies], the cross section for trapping should be quite small. Overall, the behavior that we observe agrees quite well with expectations based upon the effective mass approximation. Even though the results in Figure 9 are very reasonable we need to point out that there is an approximation involved in treating Ag0 on a lattice site as though it were a conduction electron in a shallow trapped state. The conduction electron is usually thought of as a diffuse charge in the shallow electron state. To better represent the electron, we consider a model where one electron is spread over several silver ions located on the surface of a sphere that is centered at the Fe(CN)64- complex. This is a demanding calculation, so the sphere size must be limited. The binding energy computed for this spherical method as compared to using Ag0 is shown in Table 5 for three distances. The differences in binding energy are small, and this result adds support to the approximation of using Ag0 as a representation for the shallow trapped electron for dopants of high symmetry. We note that there are some uncertainties concerning the nature of the electron in the shallow trapped state. In the effective mass approximation the electron is in a 1s orbital, which gives a maximum density at the site of the dopant that decays exponentially with distance. The radial probability requires multiplication of the density by the square of the distance, and this function has its maximum away from the dopant and near the distance assigned to the radius of the shallow electron state. The sphere method, used in the atomistic calculations, assigns all of the electron density to the surface of a sphere at the effective radius value and this also is an approximation. Thus, relative comparisons are emphasized in the calculated binding energy of the shallow trapped electron since all dopants are subjected to the same approximation. Quantum Mechanical Calculations of Fe(CN)6x- in AgCl. Embedded cluster calculations were performed for various charge states of Fe(CN)6x- with and without compensating cation vacancies in AgCl. The properties of the optimized structures of these are given in Table 6. The C-N bond length
1136 J. Phys. Chem. B, Vol. 101, No. 7, 1997
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TABLE 6: Optimized Geometry and Charge of Iron Hexacyanide Dopants in AgCl bond length dopant
vacancy
Fe(CN)64Fe(CN)63-
0 0
Fe(CN)64-
(002)
Fe(CN)64-
(110)
Fe(CN)63-
(002)
Fe(CN)64-
(002) (002h)
Fe(CN)63-
(002) (002h)
Fe(CN)64-
(110) (1h1h0)
Fe(CN)63-
(110) (1h1h0)
Mulliken charge
bond
Fe-C (Å)
Fe
C
N
eq ax eq ax axa eq ax eqa eq ax axa eq ax eq ax eq ax eq ax
2.075 1.988 1.993 2.088 2.088 2.043 2.101 2.095 2.074 1.995 2.010 1.975 2.104 2.051 2.153 2.035 2.102 2.116 2.005 2.027
1.20 1.52
-0.51 0.48 -0.49 -0.49 -0.49 -0.73 -0.49 -0.49 -0.49 -0.47 -0.47 -0.69 -0.48 -0.71 -0.48 -0.78 -0.39 -0.46 -0.38 -0.44
-0.36 -0.27 -0.27 -0.39 -0.39 -0.09 -0.40 -0.40 -0.35 -0.29 -0.29 -0.02 -0.41 -0.11 -0.31 0.02 -0.45 -0.44 -0.34 -0.35
1.21 1.19 1.53 1.21 1.66 1.16 1.47
a Bond in the direction of the vacancy where the bonds are inequivalent.
TABLE 7: Ionization Energy (eV) Values for Embedded Fe(CN)64- Dopants in AgCl
vacancies 0 (002) (110) (002) (002h) (110) (1h1h0) (110) (1h10) (200) (1h10) AgCl65-
SCF/ROHF electronic polarization IP energy
IP
SCF/ROHF MP2 IP
12.06 9.38 8.71 7.67 5.52 6.73 5.98
-5.96 -3.60 -2.13 -1.23 0.53 1.03 1.13
6.10 5.78 6.58 6.44 6.05 7.76 7.11
5.33 6.50 6.49 5.74 5.04 8.51 7.78
6.85 6.19 7.35 7.26 5.48 5.78 6.09
8.38
-1.42
6.96
6.84
6.96
local density IP
was rather insensitive to charge state or the presence of vacancies and remained within 0.01 Å of 1.15 Å so it is not reported. In the iron formal 3+ state the Fe-C bond length is shorter than in the formal 2+ state. The presence of vacancies near a CN- ligand causes a shortening of the Fe-C bond length. This result, shown for (002) and (002) (002h) vacancies, is consistent with expectations based upon Coulomb considerations since the effective negative charge of the vacancy repels the CN- ligand. In the case of the (110) (1h1h0) divacancy complex, the axial Fe-C bond length is greater than the corresponding equatorial distance for Fe(CN)63-, but the reverse is true for Fe(CN)64-. In this configuration the presence of vacancies causes a distortion like that shown in Figure 3 where the equatorial bonds are bent off-axis. A distortion of 2.2° is found in these calculations. Vacancies also cause a polarization of the nearby CNligands. The data in Table 6 show that the N nearest a vacancy becomes more positive in charge while the C becomes more negative. Charge tends to be pushed off the N onto the C for the (002) and (002) (002h) vacancies. This effect is less strong for the (110) (1h1h0) divacancies and may reflect the difference in cyanide molecular orbitals that point to the vacancy. The σ molecular orbitals lying along the bond axis point to the (200) position, while the π orbitals point to the (110) position. Thermal ionization energies of the iron hexacyanide complexes in AgCl have been calculated using Hartree-Fock with and without MP2 and the BLYP local density method described before. These values are reported in Table 7. For the SCF/
ROHF type of calculation we show the electronic IP before correction for lattice polarization and the final IP value. The polarization energy is largest for the uncompensated complex and becomes smaller in magnitude with compensation by silver ion vacancies. Clearly, the lattice polarization term is important. For the MP2 and local density calculations we only report the corrected IP. We reference these values to the top of the AgCl valence band by use of the AgCl65- embedded cluster. The IP of the Fe(CN)64- and in most cases the IP of the vacancy compensated Fe(CN)64- is less than that of AgCl65-, indicating a trap for holes on the complex. The same is true of most of the divacancy complexes. Inclusion of MP2 correlation effects in the calculation does shift the overall IP for some of the vacancy complexes, but it does not change the possibility that these could trap holes. Similar effects are noted in the local density calculation. Since the geometry was not reoptimized in the MP2 or local density calculations, some error could be introduced in our procedure. Thus, we attach greatest significance to the agreement in trend of the different calculations. We examine the possibility that Fe(CN)64- can act as a shallow or deep electron trapping center by computing its electron affinity in AgCl. We compute the value 2.51 eV using the SCF/ROHF calculations and including lattice polarization energy corrections. This value becomes 3.06 eV after including the MP2 corrections. The position of the conduction band edge in AgCl may be estimated by adding the experimental band gap of 3.2 eV to the IP of AgCl65-. This gives a value of 3.63.7 eV for the conduction band edge in AgCl. The electron affinity of Fe(CN)64- is less than this value so the complex cannot become a deep trap for electrons. It can only function as a shallow electron trap. The picture that emerges from these calculations is very consistent with one-electron models1,2 which are often applied to these d6 dopants. In this scheme the crystal field splitting of the CN- ligands is strong, causing low spin fully occupied t2g levels. The eg levels are above the t2g levels by 10 Dq or about 4.2 eV for this dopant. Thus, if the t2g levels lie within the band gap, the eg levels must be well above the conduction band edge and remain completely occupied. Experimental measurements1,2 of hole trapping at Fe(CN)64- in AgCl indicate that its t2g levels lie in the band gap. 4. Discussion We have treated the transition-metal hexacyanide complex in the silver halide crystal lattice by atomistic classical and quantum mechanical methods. In the classical calculations we found that when Fe(CN)64- is substituted for AgCl65- or AgBr65-, there is a significant amount of lattice distortion due to misfit and the different charged units involved. The net charge of the complex is delocalized as opposed to localized charged impurities which can be introduced in silver halide. This charge results in an ability to binding a silver ion vacancy. In the Frenkel defect material silver halide, such vacancies are present through thermal processes. Our classical simulations also showed that the Fe(CN)64- complex can be a shallow electron trap if it is not associated with vacancies. We simulated a conduction band electron crudely as an electron localized on a silver ion lattice site, yet this gave a binding energy to the complex very similar to values calculated with the effective mass approximation. Two cation vacancies can associate with Fe(CN)63- in NaCl and silver halide crystals. The preferred divacancy orientation (110) (1h1h0), which we computed in NaCl, agreed well with the experimental structure derived from ENDOR.4 In this geometry
Iron Hexacyanide in Silver Halide
J. Phys. Chem. B, Vol. 101, No. 7, 1997 1137
TABLE 8: Parameters of the Calculation Intermolecular Terms D (eV)
β (Å-1)
r10 (Å)
Fe -C Fe3+-C C-N
3.0435 3.0435 9.261
1.5779 1.3312 2.2559
2.184 2.084 1.142
bond angle
K(Fe2+)
K(Fe3+)
θ (deg)
Fe-C-N C-Fe-C
2.453 0.993
2.054 0.904
180 180
interaction 2+
Core Charges 1.2236 Fe3+ -0.5353 C -0.3353 N
Fe2+ C N
1.7475 -0.4177 -0.37355
TABLE 9: Born-Mayer Interactions interaction
A (eV)
F (Å)
interaction
A (eV)
F (Å)
C-Ag+ N-Ag+ C-ClN-ClC-Br-
617.37 715.06 249.25 288.44 479.70
0.293 77 0.292 54 0.318 86 0.316 99 0.291 81
N-BrC-Na+ N-Na+ C-C N-N
554.24 176.73 207.25 91.85 127.75
0.292 19 0.320 47 0.317 18 0.366 02 0.351 44
most important effect which must be treated was shown to be the lattice polarization. This is treated in a simplified manner using a two-stage procedure. In spite of this we were able to confirm the lack of deep electron trapping levels of Fe(CN)64in AgCl. Thus, the dopant can only act as a shallow trap for electrons. The calculations showed that the presence of vacancies significantly polarized the charge distribution in the cyanide ligands of Fe(CN)64-. This effect is consistent with our classical atomistic calculations that showed the need for treating CN- polarizability in order to correctly predict vacancy configurations. We have computed that the Fe(CN)64- complex in AgCl has an IP less than the corresponding value for AgCl. This indicates that the complex, free or associated with one vacancy, can trap holes. In an one-electron picture the octahedral Fe(CN)64complex has three doubly occupied t2g orbitals. Our calculations predict that these t2g levels lie 1.18 or 0.38 eV above the AgCl valence band edge for Fe(CN)64-V002 and Fe(CN)64-V110, respectively. The calculations are also consistent with hole trapping when two vacancies have become associated with the complex. These effects seem to be in agreement with experimental observations.1,2
TABLE 10: Buckingham Interaction N-Ag π N-Ag+ σ +
A (eV)
F (Å)
C (eV Å6)
4461.54 9729.68
0.29051 0.23800
197.41 57.51
the vacancies force the cyanide ligands to bend away causing a loss of octahedral symmetry. More complicated divacancy complexes were predicted in AgCl and AgBr. In this case the CN- ligand polarizability leads to favored orientations such as (2h00) (020); (200) (200); and (1h1h0) (200). Ligand polarizability is introduced through the shell model, but there is not an independent means of specifying this polarizability value in the silver halide lattice. Our calculations over a range of plausible polarization values for CN- led to predictions in general agreement with inferences from EPR experiments.1,2 We have found that the CN- ligand has a bonding interaction with neighboring silver ions in the silver halide crystal. Inclusion of this interaction in the classical atomistic calculations is necessary in order to explain the favored (200) vacancy position in AgCl and the (200) (2h00) divacancy positions reported in experiments. It is reasonable that the nitrogen lone pair electrons interact with silver ions since the empty 5s silver orbital should accept some electron density. This type of covalent bonding between the cyanide ligands and the host silver ions can provide a very efficient coupling of dopant to the host lattice. Silver ions at (110) positions have twice as many Ag+-N nearest-neighbor interactions than silver ions at (200) positions. Thus, for bonding interactions this tends to favor formation of (200)-type vacancies in preference to (110). There are clearly some deficiencies in the classical simulations. This involves a lack of procedures for obtaining charge self-consistency between the crystal lattice and the transition metal complex. A good example involves the charge rearrangement induced by the presence of vacancies as shown for Fe(CN)64- in Table 6. Likewise, there is no good means of independently specifying the CN- polarizability independent of calculations. Despite these deficiencies a workable means of specifying the location of cation vacancies near hexacyanide dopants seems to have been achieved. We have applied a two-stage quantum mechanical and lattice polarization type of calculation to iron hexacyanide complexes in AgCl. The calculations employ rather elaborate basis functions and take into account electron correlation effects. The
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