J. Phys. Chem. B 2001, 105, 7533-7541
7533
A Computer Simulation Study of Silver-Gold Cluster Formation on AgBr Tabular Microcrystals with AgIBr Cores R. K. Hailstone* Chester F. Carlson Center for Imaging Science, Rochester Institute of Technology, 54 Lomb Memorial DriVe, Rochester, New York 14623
R. De Keyzer R&D Materials, Agfa-GeVaert Group, Septestraat 27, B2640 Mortsel, Belgium ReceiVed: March 20, 2001; In Final Form: May 14, 2001
Computer simulation is used to study how an iodobromide core in an AgBr tabular microcrystal affects the efficiency of photoinduced metal cluster formation. Iodide ions can trap photogenerated holes and act as recombination centers. But interstitial silver ion formation from lattice silver ions adjacent to the trapped holes can reduce the charge on the site, reducing the recombination cross section and increasing the overall efficiency of metal cluster formation. This process is modeled by varying the recombination radius of the iodide-trapped hole over an 8-fold range. Several arrangements of electron traps provided by a chemical treatment of the microcrystal surface are studied. The simplest arrangement is that of a uniform placement. In this case, the iodobromide core can improve efficiency but only when the recombination radius is smaller than that due to intrinsic hole traps. When the electron traps are positioned at the corners of the microcrystal and given a large trapping radius and trap depth, the iodobromide core is unable to improve the efficiency, even when its recombination radius is 25% of that for an intrinsically trapped hole. Locating some of the electron traps at the thin edges of the tabular microcrystal benefited from the iodobromide core only when the traps had the same trapping radius as those on the face of the microcrystal. Similar results were found when the electron traps were confined to a small region of the edge. Comparison with limited experimental data suggests that the interstitial-silver-ion formation process at the site of an iodide-trapped hole appears to be competitive with recombination at that site. This situation leads to an efficiency improvement in those cases where there is excessive recombination, even after optimum chemical treatment of the microcrystal surface.
Introduction Conventional photographic materials consist of silver halide microcrystals embedded in a layer of gelatin and water. Their sensitivity depends on (1) the fraction of incident light absorbed by the layer, primarily determined by the size of the microcrystal; (2) how efficiently the microcrystal uses the absorbed light to form a developable metal cluster (the photographic latent image); and (3) the threshold size of the latent image required to initiate the development reaction. In the present study, we are concerned with the second factor. The efficiency of silver cluster formation in silver halide microcrystals is very low, often requiring a hundred or more absorbed photons/microcrystal for half the microcrystals to reach the developable state.1 This situation can be improved considerably by a process called chemical sensitization, in which the microcrystals are treated with compounds containing labile sulfur and gold atomssreferred to as sulfur-plus-gold sensitization.2 The products of this reaction are not known in great detail, but are thought to be silver-gold sulfide ionic clusters attached to or imbedded into the microcrystal surface.2,3 They act as electron traps and promote the formation of the metal clusters during exposure.4 Under optimum conditions, the efficiency * To whom correspondence should be addressed. email: hailstone@ cis.rit.edu.
improves to the point that only 8-12 absorbed photons/ microcrystal are needed to make half the microcrystals developable.5 For optimal development conditions, the minimum developable size of the latent image in the case of sulfur-plus-gold sensitization is three atoms,5,6 being an unknown ratio of silver to gold atoms. If the photographic system worked at full efficiency, then only three absorbed photons would be needed to convert a microcrystal to the developable state. Thus, there is room for considerable improvement in efficiency even in the case of optimum sulfur-plus-gold sensitization. Such an improvement is an attractive option because highly efficient microcrystals enlarge photographic space, allowing more degrees of freedom in system design and optimization. The primary limiting factor in achieving highly efficient latent-image formation is the recombination between photogenerated electrons and holes.5,7 Although sulfur-plus-gold sensitization mitigates this recombination to a large extent by providing an efficient pathway for metallic cluster formation, it is unable to eliminate it entirely. There are several strategies for improving this situation.7,8 In this study, we focus our attention on one of them, separating the electron and hole by building a heterojunction into the microcrystal structure.8b-g By building in an iodobromide core into the AgBr microcrystal, a mismatch of the valence bands in the core and shell
10.1021/jp0110511 CCC: $20.00 © 2001 American Chemical Society Published on Web 07/18/2001
7534 J. Phys. Chem. B, Vol. 105, No. 31, 2001
Hailstone and De Keyzer
Figure 1. Model space used for the simulations defining the terms used in the text.
region leads to a confinement of the photogenerated holes in the core region.9 Because the silver-gold sulfide clusters are at the surface of the microcrystal, there is a possible electronhole separation mechanism that might lead to higher efficiencies. The success of the strategy relies on a number of parameters, only some of which are known. We investigate the success of such a strategy with a computer-aided design tool10 which allows us to determine the critical factors related to a realized efficiency improvement. In the next section, we describe the simulation approach, the microcrystal architecture employed, and the parameter values. Then, we discuss in detail how we treat the iodobromide core and select parameters to be studied. Next, we discuss the simulation results showing that an efficiency improvement is realized only if the recombination radius of iodide-trapped holes is smaller than that for surface-trapped holes, and even then, only when the surface electron trapping radius is at its nominal value. A comparison with limited experimental data is used to assess the appropriateness of chosen parameter values. Finally, we include a summary of the important findings of this work. Computer Simulation Methodology Simulation Description. Details concerning the simulation program can be found elsewhere.10a Here, we give a brief review. The simulation uses a Monte Carlo approach to follow the absorption of photons, movement of charge carriers, trapping and detrapping of electrons and holes, formation and decay of silver atoms, nucleation of the silver or silver-gold clusters (two-atom) and their growth to a developable size, and recombination between electrons and holes. These processes are all carried out within a model space meant to simulate the size and shape of the microcrystals being studied. The simulation is initiated by defining the length, width, and height of the model space, a typical example for a tabular microcrystal being shown in Figure 1. Also, the parameters defining any interior boundaries, such as iodobromide regions, can be specified. Then, the trap density and location associated with the silver-gold sulfide clusters must be specified, as well as their depth and trapping radius. In the present case, where tabular grains are of interest, we differentiate between traps located on the large faces and those on the edge by using different trapping parameters for these two locations. In addition, the simulation output is designed to monitor the metal cluster formation at each location. Trapping sites on silver halide microcrystal surfaces are thought to be associated with the kink sites that are inherent to ionic crystal surfaces.11 Using a crystal with {100} surfaces as an example, these kink sites are formed at the terminus of an incomplete step, as shown in Figure 2. On {111} surfaces, where the crystallographic nature of the reconstructed surface is less clear,12 we use the term kink-like site. The main point is that these are sites of uncompensated charge which formally is (0.5e because the ion in question is missing three of its normal six nearest neighbors. A second-order refinement would take into account relaxation by the surrounding ions to minimize the
Figure 2. Illustration of a kink site on a {100} surface of a facecentered cubic ionic crystal. Illustration shows a positive kink which would act as an electron trap. Interchanging the charges would produce a negative kink, which would be capable of trapping holes. These sites typically have small binding energies (trap depths) for the carrier because of the high dielectric constant of the crystal. However, holes are more deeply trapped than electrons because of the higher effective mass of the hole.
excess charge. The charge on the kink site would also be affected by the vibrational properties of the surface and near-surface ions. We neglect this effect and consider only the time-averaged charge. The silver-gold sulfide clusters are thought to form near or at the positive kink sites,13 where the excess charge enhances the electron trapping radius and provides a greater trap depth. Photon absorption can be either a volume or surface process, depending on the problem under study. The latter case refers to an adsorbed dye used to extend the absorption beyond that possible with the blue-absorbing silver halide microcrystal.14 For either situation, the location of the photon absorption is randomly chosen and it is at this point that the random walk of the electron and hole begins. Input parameters specify the mean distance per jump of the carriers. The distribution of actual jump distances follows that of a Gaussian distribution. In the present study, we are concerned with the undyed case. This provides a simplification because we do not have to follow possible holetrapping by adsorbed dye.14 In principle, after each carrier jump, the possibility of trapping or recombination should be evaluated. But, partitioning of the grain into regions allows for more efficient simulations because checks for trapping or recombination need only be done in those regions where there is a finite probability of the event occurring. A trapping probability is calculated based on the density of traps and their cross section. When the electron passes close enough to or through a region of traps, it is trapped based on this probability and a selected random number. In the case of hole trapping at the surface, the density of hole traps (negative kink or kink-like sites) is assumed to be high enough that there is unit probability of trapping. Hole trapping at iodide sites is treated in a later section. Upon trapping, the lifetime of the carrier in the trap is determined by the trap depth and the attempt-to-escape frequency. Also, upon electron trapping, an atom formation time is determined. This value is correlated with the ionic conductivity of the microcrystal. In the case of sulfur-plus-gold sensitization, the atom state could be a gold atom. If the atom formation time is shorter than the trap escape time, then atom formation will be favored. Otherwise, electron escape to the conduction band will dominate. The time for both events is determined by an exponential probability distribution. Upon atom formation, an atom lifetime must be calculated. Decay of the silver atom competes with electron trapping to form a two-atom center, the nucleation event. For formation of two-atom and larger centers,
Silver-Gold Cluster Formation
J. Phys. Chem. B, Vol. 105, No. 31, 2001 7535
Figure 3. Energy-band diagram used to illustrate the reversible processes involving the electron that are followed in the simulations. Interstitial silver ion symbolized as Agi+. In the case of sulfur-plusgold sensitization, the atom state may be a gold atom.
the capture of the interstitial silver ion is assumed to be fast and is not explicitly followed. Some of these concepts are illustrated schematically in Figure 3. There are two pathways for recombinationsbetween free electrons and trapped holes and between free holes and trapped electrons. Usually, the former pathway dominates because holes, with their higher effective mass, are more deeply trapped.11 In addition, the density of traps for holes appears to be higher than for electrons.11 However, upon chemical sensitization electron traps with larger trap depths than those provided by the intrinsic lattice defects are formed,15 making this recombination pathway less competitive with metal cluster formation. At some trap depth/density combination, the efficiency decreases because free-hole/trapped-electron recombination dominates. Optimum chemical sensitization is achieved when the probability of both these recombination pathways is minimized.16 When all the electrons are consumed by the irreversible processes of nucleation, growth, or recombination, the simulation of one microcrystal is completed. However, because all the processes are random, the final state of that microcrystal may or may not be that of the mean response of an ensemble of microcrystals. So, the entire simulation process must be repeated many times, each with a different seed for the random number generator to achieve a mean response that can be compared with experimental data. For this reason, simulation on a parallel processor is highly advantageous.17 Following the simulation over the specified number of “microcrystals,” the fraction F made developable is determined. Then, the photon level is incremented and the entire process repeated to eventually achieve an F-log P curve, where P is the mean number of absorbed photons/microcrystal. The simulated efficiency results in this paper will be presented as quantum sensitivity values. Quantum sensitivity (QS) is defined as the mean number of absorbed photons/microcrystal to make half of the microcrystals developable. This is a common method for evaluating the efficiency of developable metal cluster formation in photographic materials but it has the unfortunate property of causing some confusion to those not in the field. A higher efficiency will translate into a lower QS value, and vice versa. The highest efficiency possible for the conditions studied in this report is three absorbed photons/microcrystal. All results will also pertain to a 0.1 s exposure time, the results at 10-6 s being very similar. Validation of the simulation approach described above,10 and its simpler versions,6,16,18 have been demonstrated for threedimensional grains. A similar demonstration for tabular microcrystals is not possible because so little data has been published regarding their fundamental sensitometric properties. Moreover, fabrication of tabular microcrystals often involves more variables than in three-dimensional microcrystals that can have significant
Figure 4. Electron micrograph of carbon replicas of tabular microcrystals. Magnification ∼10 000×. Shadows are due to evaporation of metal at an angle and can be used to assess the thickness of the microcrystals.
photographic consequences, making comparisons between predicted effects and experimental data difficult. Nevertheless, the present work is still deemed worthwhile as a starting point in modeling these very important commercial materials. When more fundamental experimental data are available, the simulation can be focused on the particular microcrystal design to demonstrate the model’s utility more clearly. Microcrystal Architecture. The use of tabular microcrystals in films intended for camera exposures is very common, and an electron micrograph of such microcrystals is shown in Figure 4. To achieve this anisotropic growth pattern it is necessary to form internal crystal defects called twin planes. These twin planes lead to rapid lateral growth relative to growth in the thickness direction.19 Microcrystals used in photographic products have diameters from submicron to tens of microns, and thicknesses from tens to hundreds of nanometers. For the simulations, the tabular “microcrystal” is modeled as a square slab having edge length 1 µm and thickness 0.2 µm. The microcrystal has an AgI.10Br.90 core that is 32% of the microcrystal volume. This core is covered by a shell composed of AgBr, making the overall iodide content 3.2%. A cross section of the model microcrystal is shown in Figure 5. As mentioned above, significant runtime savings can be achieved by dividing the microcrystal into regions which have common features. By simply knowing in which region the carrier currently resides the simulation program knows which events are possible and, more importantly, which are not. The companion drawing in Figure 5 shows the boundaries of these regions and Table 1 gives an explanation of what events can happen in each of the five regions. Parameters. An important concern is the choice of mean jump distance for the electron and hole. If the jump distance is very small, then the runtimes increase dramatically. If the jump distance is too large, then we loose the atomic-like nature of the simulations. Experience has indicated that using a mean jump distance that is 20% of the smallest grain dimension is a good compromise,10a which in the present case leads to a mean jump distance of 0.04 µm for the electron. Holes are less mobile than
7536 J. Phys. Chem. B, Vol. 105, No. 31, 2001
Hailstone and De Keyzer TABLE 2: Parameter Values Used in the Simulations parameter
value
time step diffusion coeff, electron diffusion coeff, hole S + Au trap depth S + Au trap radius S + Au trap density exposure time atom formation time atom decay time hole trap lifetime, neg kink hole trap depth, Io hole trap depth, I2recombination radius, neg kink recombination radius, Io no. of microcrystals
10-11 s 108 µm2/s 106 µm2/s see Table 3 see Table 3 see Table 3 0.1 s 10-7 s 10-4 s 10-9 s 0.12 eV 0.18 eV 5 nm 1.25-10 nm 400
TABLE 3: Properties of Silver-Gold Sulfide Clusters with Uniform, Face, and Corner (Edge) Locations
Figure 5. (Top) Dimensions of the tabular “microcrystal” and its iodobromide core. Units are in µm. (Bottom) An expanded view of the microcrystal showing the five regions used to follow the electrons and holes in the simulations. Events possible in each region are given in Table 1.
TABLE 1: Possible Events in Each Region of Figure 5 region
possible events
1
e- free, h+ free or trapped at I-, e- can recombine with trapped h+ h+ can escape to AgBr region, e- and h+ can recombine h+ can enter into AgIBr region e- and h+ free (no trapping or recombination possible) e- and h+ can be trapped at surface traps, e- and h+ can recombine
2 3 4 5
electrons in AgBr, having a microscopic mobility that is about 60x smaller.20 In the simulations, we make the jump distance of the hole 1/100 that of the electron. Silver halide microcrystals are characterized by an ionic space charge layer that arises from the generation of interstitial silver ions from surface positive kink sites.21 These interstitial ions migrate into the microcrystal leaving behind an excess of negative kink sites. The surface acquires a negative charge, balanced by a subsurface layer of positive interstitial silver ions. The negative surface potential causes the conduction and valence bands to bend upward at the surface, acting as a barrier for electrons and an attractive potential for holes. Current hardware limitations prevent the incorporation of these computationally challenging space charge effects into the simulation software. However, qualitative simulation studies suggest that the effects are small, primarily because, in addition to the electronic effects, there are also ionic effects to consider and the two effects tend to balance one another. This conclusion is supported by experimental data, which show that the efficiency of latent-image formation in sensitized microcrystals is the same whether the chemical sensitization is done at the surface5 or in the interior.22 In the former case, the electrons would have to overcome the surface potential and then avoid recombination with the holes that are preferentially localized at the surface by the same potential. In the latter case, latentimage formation should be able to take advantage of the builtin space-charge-induced separation of electrons and holes. Another parameter of importance in the simulation is the radius of trapping and recombination. Because the sites for trapping and recombination are assumed to have a partial
trap location
trap depth, eV
trap radius, nm
trap density, µm-2
uniform face corner(edge)
0.20 0.10 0.40
5.0 2.5 10.0
800 3000 30
electrostatic charge a radius can be derived by equating the Coulomb potential energy to the thermal energy, expressed as kT, where k is Boltzmann’s constant and T is the temperature on the Kelvin scale. For parameters appropriate to the silver halides, the radius is 2.5 nm.10a However, it turns out that the ratio of trapping to recombination radius is what determines the efficiency of latent-image formation, all other parameters being fixed. By increasing both radii to 5 nm we can achieve faster simulation times but with the same result.10a In the present study, we have also varied the trapping radius between 2.5 and 10 nm to see how different trap types might affect the efficiency of latent-image formation. The recombination radius in the iodobromide core was also varied, as described in the next section. The parameters and their values are summarized in Tables 2 and 3. Iodobromide Core Band Energies. To treat the electron and hole motion through the microcrystal we need to know how iodide alters the band structure of AgBr. From absorption studies of AgIBr, we know the presence of iodide causes a narrowing of the band gap in the iodobromide region.23 Mason, using X-ray photoelectron spectroscopy, has concluded that the narrowing of the band gap occurs almost exclusively by an upward movement of the valence band maximum.24 We will assume in the simulations that the iodide affects only the position of the valence band maximum in the core relative to the AgBr shell. The extent of the narrowing of the band gap is a function of iodide concentration but most of the decrease occurs at low iodide concentrations.25 Berry estimates from optical absorption data on bulk materials that the decrease at 5% iodide concentration is about 0.2 eV.9b We use this value in our simulations. Furthermore, because we assume that the interface between the iodobromide core and the bromide shell is an abrupt one, the 0.2 eV energy difference effectively permanently localizes any holes within the core region on the time scale of latent-image formation. Concentrations. As the concentration of the iodide in the core is increased, it becomes more likely that two iodides are on adjacent lattice positions forming iodide pairs. Because these iodide pairs have different electronic properties than isolated
Silver-Gold Cluster Formation
J. Phys. Chem. B, Vol. 105, No. 31, 2001 7537
iodides, it is important to calculate their relative number. Assume that iodide is randomly distributed on the halide sublattice, that N0 is the number of halide sites per unit volume, and that N is the number of iodide sites per unit volume. Then the probability of a halide site having an iodide is simply N/N0.26 The probability of a halide site having an iodide and also having another iodide as a nearest neighbor is
N nN × N0 N0
(1)
where n is the number of nearest neighbor halide positions in the lattice, which is 12 for the face-centered cubic lattice of AgBr. The concentration of nearest neighbor pairs is then given by
1 nN2 nN2 × N × ) 0 2 2N0 N02
(2)
where the factor 1/2 corrects for double counting. For N/N0 ) 0.10 (10% I-) the double iodide concentration will be 0.6N. That is, 60% of the iodide exists as double iodides and 40% as single iodide. In the above calculation, we have neglected higher-order clusters such as trimers and tetramers. Because they would have double iodides as part of their structure, we have simply lumped them all together and classified them, as least as far as their electronic properties, as the same species which we refer to as double iodide. A future refinement might take into account the higher-order clusters, although little seems to be known about their electronic properties. An additional simplification we have used is to assume that the pairing of iodides is only driven by statistical factors. There may also be covalent forces at work that may increase the tendency of iodides to cluster together. Trap Depths. At 10% iodide there will be about 108 I- in the iodobromide core of our model tabular microcrystal. This concentration is much too large to follow the trapping and detrapping of holes explicitly; an approximation must be used. Another consideration is how the single and double iodides will affect the trapping of holes as they move through the iodobromide region. Tsukakoshi and Kanzaki,27 as well as Czaja,28 have observed a blue shift in the 4.2 K luminescence peak in AgIBr as the iodide concentration increased. The shift could clearly be seen at 0.5% iodide and was attributed to the formation of iodide pairs. Both the single- and double-iodide peaks were quenched at higher temperatures, but with different activation energies 30 and 90 meV, respectively. The quenching was due to thermal detrapping of the electron from the exciton bound at the singleor double-iodide site. Ignoring lattice relaxation effects, which seems appropriate for the low temperature used for these studies, we can derive an approximate energy level diagram. From the thermal quenching energies, along with the peak positions of the single (495 nm) and double (520 nm) iodide, one can estimate that the depth of the trapped hole at a double iodide is 60 meV greater than at a single iodide. Furthermore, the depth of a trapped hole at a single iodide site is 170 meV. In deriving the energy level picture for single and double iodide we have neglected entropic considerations. This is the usual practice in discussions of electronic transitions in solids.29 Allowing for a decrease in the band gap on going from 4.2 K to room temperature30 would decrease the trap depths further. But, higher temperatures may allow lattice relaxation processes
to occur that may deepen the traps,11 as well as affect the recombination radius (see below). The depth of trapped holes in an AgI.05Br.95 sheet crystal has been measured as 0.15 eV.31 At this iodide concentration eq 2 indicates that 30% of the iodide is present as pairs. So, the measured trap depth might be considered as a weighted average between that for single and double iodides. On the other hand, Hamilton suggests 0.40 eV for the double iodide, based on other luminescence data.32 In the final analysis, we have chosen trap depths of 0.12 eV for a hole trapped at a single iodide and 0.18 eV for a hole trapped at a double iodide. We will see that our results are not particularly sensitive to the chosen trap depth. Hole Motion and Trapping. We assume that the hole has the same microscopic mobility in both the AgBr and AgIBr regions, but that hole trapping can occur at single and double iodides in the AgIBr region. Because both of the iodide centers are isoelectronic traps, they have no long-range Coulomb attraction for the hole and will therefore have a small trapping cross section. Given these features, the simulation code is designed so that after each hole jump in the core region a random number is generated. If this number is equal to 0.1 or less, then hole trapping occurs. Otherwise, the hole is free to make another jump. If trapping occurs, then another random number is generated. If it is equal to 0.4 or less, then hole trapping has occurred at a single iodide, otherwise hole trapping has occurred at a double iodide. Recombination Radius. Once the hole is trapped at an iodide ion the resulting Io center has a charge +e with respect to the surrounding lattice. It is, therefore, a significant recombination center since holes trapped at surface kink-like sites have a formal charge +0.5e. Because the recombination radius is proportional to the charge on the center, the iodide-trapped hole has twice the recombination radius of an intrinsically trapped hole. This statement assumes that there is no lattice relaxation occurring on the time scale of latent-image formation. The lattice relaxation could take two forms. Because of the large excess charge there would be a driving force for the surrounding lattice ions to rearrange themselves so as to minimize the excess charge. This would probably happen on a relatively fast time scale so that the effective recombination radius is smaller than that produced upon initial trapping. A second related relaxation mechanism is the formation of an interstitial silver ion. The Io may cause an adjacent lattice Ag+ to move into an interstitial position, creating a vacancy whose negative charge exactly compensates the excess positive charge introduced by the formation of Io.33 Mitchell assumes the formation of the interstitial ion is fast (picosecond time scale) because the Madelung energy at the Ag+ is reduced by more than half by the formation of Io.34 Hamilton has suggested that the lattice silver ion in question should not be viewed as a normal lattice ion, but rather as one associated with a lattice defect, analogous to a silver ion at a positive kink site (see Figure 2).11 The free energy of formation of interstitial silver ions ∆Gi at positive kink sites has been estimated to be in the range 0.27 to 0.33 eV.35 The time scale τ for formation of the interstitial would be given by
τ ) (10-12) exp
∆Gi kT
(3)
The preexponential factor is the inverse of the attempt-to-escape frequency, equivalent to the phonon frequency. Equation 3 yields a formation time of about 50-500 ns for the above free energies. Baetzold36 has calculated the free energy of formation of an
7538 J. Phys. Chem. B, Vol. 105, No. 31, 2001
Figure 6. Quantum sensitivity (QS) vs Io recombination radius for uniformly distributed S + Au electron traps. Reference line for the AgBr result uses 5 nm recombination radius for a hole trapped at a negative kink. The simulated QS for AgIBr/AgBr uses both the 5 nm recombination radius of the surface-trapped hole and the indicated Io radius. The theoretical limit is based on a minimum developable size of three atoms, which leads to a QS of three absorbed photons/grain if there are no inefficiencies.
interstitial from a positive kink site as 0.22 eV and eq 3 yields a formation time of about 5 ns. These times are considerably longer than that estimated by Mitchell. But, more importantly, the energies involved are greater than the estimated 0.12 to 0.18 eV trap depth for the single and double iodides, suggesting that the lattice relaxation by the interstitial formation process will not compete well with hole detrapping. In addition, unlike formation of the interstitial at a surface kink site where a vacancy is not formed in the process, formation of an interstitial at an iodide-trapped hole site does involve vacancy formation. So, one might want to use a higher free energy of formation than used in the above calculations. As should be evident from the preceding discussion, the time scale for lattice relaxation at an iodide-trapped hole is quite uncertain. Rather than investigate a range of possible relaxation times, we have chosen to make the recombination radius of the iodide-trapped hole a variable in our simulations. At the upper end, we use a value of 10 nm, which is twice that for a hole trapped at a negative kink at the surface and reflects the 2x difference in charge of the trapped-hole state. This recombination radius assumes lattice relaxation does not compete with hole detrapping. At the lower end, we use a value of 1.25 nm, which would correspond to an effective charge of +0.125e, designed to mimic rapid interstitial ion formation. Results Uniform Trap Location. This is the simplest case in which all the traps created by the chemical sensitization process are uniformly distributed on all surfaces of the microcrystal. The electronic properties of the silver-gold sulfide clusters are the same for face and edge locations and are given in the first row of Table 3. In earlier simulation work, these parameter values enabled highly efficient latent-image formation for threedimensional microcrystals having a volume similar to the one in the present study.10b The results are shown as quantum sensitivity vs Io recombination radius in Figure 6, with the results for the AgBr case included as a reference line. In the AgBr case, the only hole traps are surface negative kinks, which have a recombination
Hailstone and De Keyzer
Figure 7. Same as Figure 6 except the surface electron traps have a bimodal distribution as indicated in the second and third row of Table 3. In this case the highly effective traps are located at the eight corners.
radius of 5 nm. These results show that the iodide core decreases efficiency of latent-image formation when the Io recombination is greater than that for recombination at surface negative kinks, but that it improves efficiency when it is appreciably less than that of surface negative kinks. At the highest recombination radius of 10 nm, the results for the iodobromide core show that less than 4% of the holes localized in the core region eventually escape and that more than 98% of the recombination occurs within this region. Even at the lowest iodine recombination radius 95% of the recombination occurs in the core region. These results are expected based on the large barrier for hole escape from the iodobromide core. We also varied the hole trap depth (both single and double iodides) to 0.06 eV lower and also 0.06 eV higher than that chosen in the previous section. As might be expected, the efficiency was better with smaller traps depths and worse with larger trap depths. These changes were only seen at the larger Io radii. This exercise was somewhat unfair since we did not study the interstitial formation explicitly. If we had, then these hole trap depth changes would have produced even smaller effects since the deeper traps would have allowed more time for interstitial silver ion formation, decreasing the recombination radius of Io, and similarly, but in the opposite direction, for smaller trap depths. These results serve to emphasize that the chosen trap depths are not critical to the outcome of the simulation. Corner Trap Location. This is the first of the spatially distinct bimodal trap distributions. It is meant to be a crude approximation to the situation where epitaxial depositions have been applied to the corners of tabular microcrystals.8e,37 The traps formed here during chemical sensitization are modeled as being more efficacious than those formed on the large faces of the microcrystal. This hypothesis follows from the known ability of chemical sensitization to direct the formation of the latent image to the epitaxial sites.37 The properties of the face and corner traps are summarized in Table 3. The portion of the total microcrystal surface area that has the corner properties is 1.7%. The results in Figure 7 show that the efficiency of the AgBr reference case has increased relative to the uniform case in Figure 6 and that now there is no advantage of the iodobromide core, even at the smallest Io radius. As might be expected from the large difference in trapping properties between corner and
Silver-Gold Cluster Formation
Figure 8. Same as Figure 6 except the surface electron traps have a bimodal distribution. In this case the highly effective traps (“edge” row in Table 3) are located along the microcrystal edges (see Figure 1), whereas the less effective traps are present on the faces (second row of Table 3).
face location, the latent image is exclusively located at the corners in all cases studied. Edge Trap Location. This is the second of the bimodal trap distributions. In this case, the entire “edge” of the microcrystal is populated with silver-gold sulfide clusters having the properties of edge traps listed in Table 3, whereas the two large faces have traps whose electronic properties are given in the second row of this table. The motivation for this exercise is the observation that the latent image in practical tabular microcrystals tends to form at the edges.11 This is apparently caused by the presence of adsorbed dye during the chemical sensitization, which seems to cause the more efficient traps to form at the microcrystal edges. Figure 8 shows that the efficiency of the AgBr reference has increased somewhat relative to the previous case in which the traps were confined to the corners. The iodobromide core simply reduces efficiency at all Io recombination radii studied. The high efficiency of the reference is due to the high proportion of microcrystal surface area devoted to the highly effective traps, which compete favorably with recombination and eliminate any possible efficiency advantage of the iodobromide core. The latent image is located exclusively at the edge in all cases. A variant of the above condition was to make the trapping radii for the face and edge traps the same (5 nm). This has the effect of making the face traps more competitive with the edge traps, but also making recombination more competitive with edge trapping (both trapping sites and the surface recombination site now all have the same radius). Figure 9 shows that the AgBr reference is now less efficient than in Figure 8 and that the iodobromide core is now able to increase the efficiency at small Io recombination radii. Note the highest efficiency attainable in the iodobromide core case is the same as that shown in Figure 8. Unlike the case above with more efficacious edge traps, a minor component of the latent image forms at the face traps. Twin Plane Trap Location. This is the final bimodal trap distribution study. The hypothesis here is that the twin planes needed for tabular crystal growth emerge at the edges and some have suggested these sites provide particularly reactive locations during chemical sensitization, leading to very effective traps.38 The twin plane separation is assumed to be 20 nm and the edge traps are located within these boundaries centered within the edge itself, having the properties ascribed to “edge” traps in Table 3. Elsewhere on the edge and on the faces the traps have the properties ascribed to face traps in Table 3.
J. Phys. Chem. B, Vol. 105, No. 31, 2001 7539
Figure 9. Same as Figure 8, except the edge and face traps have the same trapping radii (5 nm).
Figure 10. Same as Figure 6 except the surface electron traps have a bimodal distribution. In this case, the highly effective traps (“edge” row in Table 3) are located along the center of the microcrystal edges (see Figure 1) in a 20 nm-wide band, whereas the less effective traps are present elsewhere on the edges, as well as on the faces (second row of Table 3).
The results in Figure 10 show that the AgBr control is less efficient than seen in Figure 8. This would be expected because the proportion of microcrystal surface area devoted to these highly effective traps has been reduced to less than 3%, making recombination more competitive. Nevertheless, the iodobromide core is unable to improve efficiency, even at the smaller Io recombination radii. Despite the less favorable condition for trapping in this case, the latent image is predominantly located between the emerging twin planes. As in the previous section we also studied the case where the face and edge traps have the same trapping radius (5 nm). As might be expected, the data in Figure 11 show that the iodobromide core is now able to improve the efficiency when the Io recombination radius is small. Again, like the edge case, a minor component of the silver-gold clusters form at the face traps. Discussion For the parameter conditions used in this study, the incorporation of an iodobromide core in a AgBr tabular microcrystal did not improve the efficiency of latent-image formation unless the recombination radius of the iodide-trapped hole was made very small. This suggests that the formation of an interstitial silver ion from a lattice silver ion adjacent to the Io must be
7540 J. Phys. Chem. B, Vol. 105, No. 31, 2001
Hailstone and De Keyzer if generally true, they would possess higher degrees of recombination and possibly benefit more from the iodobromide core than smaller microcrystals. Surprisingly, there was no efficiency advantage for smaller microcrystals having a volume similar to or less than the model microcrystal used in the computer simulation study. But, importantly, there was no efficiency loss with the iodobromide cores, unlike some of the computer simulation results observed in our work. This comparison suggests that the interstitial silver ion formation is probably competitive with recombination at Io, and that the effective recombination radius of the iodidetrapped hole is comparable to that for a surface-trapped hole. Thus, the parameters affecting interstitial silver ion formation adjacent to Io seem to be more favorable than some of the estimated values discussed above.
Figure 11. Same as Figure 10, except the edge and face traps have the same trapping radii (5 nm).
fast in order for the iodobromide core to have a beneficial effect on quantum sensitivity. Otherwise, there is a decrease in efficiency due to the enhanced recombination within the core. When spatially distinct trapping properties were present the iodobromide core only provided an advantage in efficiency if the two types of traps had the same trapping radius, and, even then, only when the Io recombination radius was small. The purpose of the iodobromide core is to create a heterojunction in the microcrystal that acts similarly to a pn junction in semiconductors. By spatially separating the electron and hole, recombination is minimized and efficiency increases. But this crystal design only does half the job because the electron is still free to sample the entire grain. This problem was readily apparent for those cases in which the Io radius was large. There, even though the holes were highly localized within the iodobromide core, the efficiency actually decreased because electrons could freely diffuse through the iodobromide core and undergo recombination. This was true even when the surface traps were optimized for high efficiency (large trapping radius and trap depth). The only report in the literature for comparison with experimental data used three-dimensional microcrystals with 20% I- cores.8b Compared to AgBr microcrystals, an efficiency advantage was found for large microcrystals (larger than about 1 µm diameter). Large microcrystals are known to form silvergold clusters less efficiently than smaller ones.39 There is not general agreement on the reason for this behavior, but one possibility is that larger microcrystals may have a higher propensity for recombination.10d,39g This is thought to arise because the larger microcrystals have more internal lattice defects that might trap holes and enhance recombination. Thus, the high-iodobromide core is able to generate an efficiency improvement in these larger microcrystals because such microcrystals may have higher degrees of recombination than those of smaller ones. An alternative possibility is that the larger microcrystals are less than optimally sensitized because of a fog limitation. Fog occurs when the microcrystal develops without exposure and one of its many causes is the formation of overly large silvergold sulfide clusters that are able to act as latent images in terms of accepting electrons from the developer.2a Because fog is an unwanted property, the sensitization needs to be adjusted to avoid its occurrence. Thus, fewer traps than desirable are used and the extent of recombination is higher than would be the case if fog were not a limitation. There is some evidence that the tendency for fog is greater in larger microcrystals39b,c and,
Conclusions An AgI.10Br.90 core in an AgBr tabular microcrystal is able to effectively localize photogenerated holes. However, its ability to provide an increase in the efficiency of silver-gold cluster formation depends on how fast an interstitial silver ion is formed from a lattice silver ion adjacent to an iodide-trapped hole. This variable is modeled by adjusting the Io radius over an 8-fold range. For small recombination radii, there is an efficiency increase, except in the case of highly efficacious surface electron traps where the efficiency is already high. In those cases, iodide can actually decrease efficiency even when the Io radius is 25% of that for holes trapped at surface negative kinks. Comparison with experimental data suggests that the interstitial silver ion formation is fast, but not so fast as to decrease the effective Io recombination radius to zero. However, in the case of large microcrystals which may suffer from excessive recombination, even the less than hoped for maximum effect of iodide (instantaneously zero recombination radius upon hole trapping) seems to provide a decrease in recombination and an efficiency advantage. References and Notes (1) Spencer, H. E. Photogr. Sci. Eng. 1971, 15, 468. Spencer, H. E. J. Photogr. Sci. 1976, 24, 34. (2) (a) Tani, T. Photographic SensitiVity; Oxford University Press: Oxford, 1995; Chapter 6 and references therein. (b) Harbison, J. M.; Spencer, H. E. In The Theory of the Photographic Process, 4th ed.; James, T. H., Ed.; Macmillan: New York, 1977; pp 151-152. (3) Tani, T. J. Imaging Sci. Technol. 1995, 39, 386. Kanzaki, H.; Tadakuma, Y. J. Phys. Chem. Solids 1997, 58, 221. Tani, T. J. Imaging Sci. Technol. 1998, 42, 135. Malik, A.-S.; Blair, J. T.; Bernett, W. A.; DiSalvo, F. J.; Hoffmann, R. J. Solid State Chemistry, 1999, 146, 516. Charlier, E.; Van Doorselaer, M. K.; Gijbels, R.; De Keyzer, R.; Guens, I. J. Imaging Sci. Technol. 2000, 44, 235. Tani, T.; Yoshida, Y. J. Imaging Sci. Technol. 2000, 44, 242. Hailstone, R. K.; Zhao, T.; DiFrancesco, A. G.; Tyne, M. J. Imaging Sci. Technol. 2001, 45, 76. (4) Moisar, E. Photogr. Sci. Eng. 1981, 25, 45. (5) Hailstone, R. K.; Liebert, N. B.; Levy, M.; McCleary, R. T.; Girolmo, S. R.; Jeanmaire, D. L.; Boda, C. R. J. Imaging Sci. 1988, 32, 113, and references therein. (6) Hailstone, R. K.; Hamilton, J. F. J. Imaging Sci. 1985, 29, 125. (7) Babcock, T. A.; James, T. H. J. Photogr. Sci. 1976, 24, 19. James, T. H. Academy of Science of the USSR Symposium on The Nature of the Latent Image in SilVer Halide and Other Semiconductors, Moscow, May 16-20, 1977. DiFrancesco, A. G.; Tyne, M.; Pryor, C.; Hailstone, R. K. J. Imaging Sci. Technol. 1996, 40, 576. Belloni, J.; Treguer, M.; Remita, H.; De Keyzer, R. Nature 1999, 402, 865. (8) (a) Gould, I. R.; Lenhard, J. R.; Muenter, A. A.; Godleski, S. A.; Farid, S. J. Am. Chem. Soc. 2000, 122, 11 934. (b) Takada, S.; Ayato, H.; Ishimaru, S. International Congress of Photographic Science, Cologne, September 1986; 81. (c) Nakayama, T.; Ohtani, H.; Matsusaka, S. International East-West Symposium II, Kona, Hi, November, 1988; p C-85.
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