Computer Simulation Studies of Silver Cluster Formation on AgBr

4414

J. Phys. Chem. 1995,99, 4414-4428

Computer Simulation Studies of Silver Cluster Formation on AgBr Microcrystals R. K. Hailstone Center for Imaging Science, Rochester Institute of Technology, 54 Lamb Memorial Drive, Rochester, New York 14623-5604 Received: July 28, 1994; In Final Form: December 1, 1994@

Photoinduced silver cluster formation on AgBr microcrystals is modeled by a nucleation-and-growth process in competition with electron-hole recombination. Nucleation involves the trapping of an electron at a silver atom, followed by the migration of an interstitial silver ion to yield a stable two-atom cluster. The growth stage is the enlargement of the two-atom cluster by additional trapping of electrons and capture of interstitial silver ions to yield the photographically developable latent image. The simulation of silver cluster formation is accomplished with a Monte Carlo procedure which randomizes the sequence of events but weights each event according to a probability determined by the simulation parameters. Events followed are photon absorption, trapping and detrapping of electrons and holes, silver atom formation and decay, nucleation, growth, and recombination. Prior to the simulations, the microcrystal size and shape must be specified, along with the density, depth, and trapping radii of electron traps. Other parameters include the diffusion coefficient of the electron and hole, the recombination radius, the time for capture of interstitial silver ions by trapped electrons, and the lifetime of the silver atom. Simulation over an ensemble of independent microcrystals is done with a transputer-based parallel processor. Exemplary results showing the dependence of silver cluster formation efficiency on trap depth and trap density as produced by chemical pretreatment of the microcrystals are included. For the high-irradiance condition studied, the trends show that as the cluster size increases, the required trap depth and trap density for maximum formation efficiency decrease.

Introduction Photoproduced silver clusters are a critical component of the conventional photographic process. When the cluster contains at least the minimum number of atoms required to initiate the photographic development reaction, it is called a “latent image center”. The quantum yield of these clusters and the minimum number of atoms required to catalyze the development reaction have a direct bearing on the sensitivity of the film layer.’ Thus, the details of the solid-state photochemical events taking place in AgBr microcrystals are of great importance in designing photographic films of increased sensitivity.2 Unfortunately, definitive experiments on AgBr microcrystals imbedded in a gelatidwater film layer characteristic of photographic film are very difficult to design. Although photographic scientists have been very clever in devising techniques to measure important physical properties of AgBr microcrystals, the relevance of these measurements to the photographic process is not often obvious. Hence, there is a critical need for a tool to connect the physics of silver halides with their photographic properties. The computer simulation tool described in this article is designed to fulfill this need. There is a good analogy between this computer simulation approach to understanding the photographic properties of silver halides and the computer-aided design of semiconductor devices. In the latter case there is a large database of physical properties of silicon and its doped versions. Models involving electron and hole diffusion can be readily constructed and used to design devices with desired electronic proper tie^.^ The goal is similar in the case of silver halides, but the database of physical properties is not as extensive. Nevertheless, even a partial achievement of the goal would provide a pleasant alternative to the largely empirical procedures now used to design new photographic materials. @

Abstract published in Advance ACS Abstracts, March 1, 1995.

0022-365419512099-4414$09.00/0

There is one important difference, however, between simulations of semiconductor device operation and photoinduced silver cluster formation on AgBr microcrystals. Each semiconductor device is expected to operate in the same way as all other devices constructed with the same techniques. Thus, the focus of the simulation is on one exemplary device, and all the computational resources can be applied to the one case under study. In photographic film layers, the photochemistry occurring in individual silver halide microcrystals is unaffected by events in the other microcrystals. Both the number of absorbed photons and the precise sequence of events occurring after light absorption are random proce~ses.~ Thus, the mean response of a collection of microcrystals, which would represent the macroscopic image, cannot be predicted from a deterministic simulation of the silver cluster formation kinetics in an individual microcrystal. Rather, a statistical response must be obtained for an ensemble of microcrystals and compared with experimental responses to gauge the correctness of the simulation~.~ The electronic and ionic processes occumng in a film layer during exposure to light are highly spatially inhomogeneous-both with respect to individual microcrystals as well as within a given microcrystal. Thus, a differential equation approach is not appropriate for simulating silver cluster formation. We use a Monte Carlo procedure to carry out stochastic simulations of the events leading to silver cluster formation. The correctness of this approach was recognized many years ago4 and has evolved to the simulation procedures discussed in this article. The mean response of the film layer is derived from repeated simulations over an ensemble of microcrystals, with a different initial seed for the random number generator within each microcrystal simulation. In the following section I discuss methods for characterizing the response of AgBr microcrystals to photons. Next, I describe the model which is used to simulate the silver cluster formation. 0 1995 American Chemical Society

Ag Cluster Formation on AgBr Microcrystals Then, I discuss how this model is implemented in our computer simulation studies. Some examples of typical results obtained from these simulations are also included. Finally, I discuss the significance of the simulations in improving our understanding of the important events in the photographic process. I also discuss future directions for the computer simulation studies.

Characterization Methods As indicated above, the general approach is to treat the microcrystals as independent units during light exposure. The light absorption within a gelatin film layer containing randomly dispersed AgBr microcrystals is assumed to obey Poisson statistics. After photographic development, the change in optical density in a given area of the film layer is a function of the number of microcrystals that have catalyzed the development reaction. The fraction of microcrystals that catalyze the development reaction is determined by the number of silver atoms present in the photoproduced clusters. Since an entire microcrystal will be converted to metallic silver if it has at least one silver cluster with at least the minimum number of atoms, the film response is often characterized by plotting the fraction of microcrystals with one or more developable silver clusters vs the log of the mean number of absorbed photons per microcrystal. The resulting sigmoidalshaped curve is known as a characteristic curve for the film layer and is one of the ways in which simulations and experiments can be compared. Another theory vs experiment comparison relies on the efficiency of silver cluster formation. This efficiency is expressed as the mean number of absorbed photons required to cause half the microcrystals in a film layer to possess at least one latent image center and therefore be capable of initiating development. The term “quantum sensitivity” is used to express this efficiency, which is in units of mean absorbed photons per microcrystal. Techniques for experimental measurement of quantum sensitivity have been described in detail el~ewhere.~ The number of silver clusters produced on the surface of a microcrystal also is important. For example, suppose the development reaction is catalyzed by a five-atom silver cluster, but a microcrystal contains a two-atom and a three-atom silver cluster. In this situation the microcrystal is not capable of initiating development even though it contains the requisite total number of photoproduced silver atoms. This situation represents an inefficiency. Plots of cluster distribution among microcrystals are useful in analyzing for this inefficiency and are another way of characterizing silver cluster formation.

J. Phys. Chem., Vol. 99, No. 13, 1995 4415

Silver halides are characterized by a high concentration of mobile interstitial silver ions.’ If a migrating interstitial silver ion should encounter a trapped electron, a silver atom forms. However, silver atoms are unstable with lifetimes probably less than 1 s.’ Thus, the initial steps of silver cluster formation are highly reversible, with many instances of electron trapping and detrapping and silver atom formation and decay. If, while one electron exists as a silver atom, a second electron should be trapped at the same site and a subsequent capture of a mobile interstitial silver ion occurs, a two-atom silver cluster forms. The two-atom silver cluster is considered to be permanently stable.’ The transition from the single-atom to the two-atom state represents a critical step in silver cluster formation and is called the “nucleation” stage. In contrast with trapping at empty traps, electron trapping at silver atoms and silver clusters is assumed to be irreversible. Although stable, this two-atom cluster cannot initiate the development reaction.8 Thus, if this center is to initiate development, additional silver atoms must form at this site by the alternate addition of electrons and interstitial silver ions. These additional events are called the “growth’ steps. The number of growth events required to produce a silver cluster that will initiate development depends on the developer and the development condition^,^ but is typically between two and four. There are many pathways by which electrons may be lost and inefficiencies in silver cluster formation arise, but the principal one is recombination. As mentioned above, free electrons and free holes cannot recombine. However, if one of these carriers is trapped and the free complementary carrier should encounter it, recombination can occur. This leads to two possible recombination pathways: free electrodtrapped hole and free holehapped electron. This model of reversible initial stages of electron trapping/ detrapping and atom formatioddecay and irreversible stages of nucleation and growth in competition with recombination has come to be called the “nucleation-and-growth model” of latentimage formation.I0 It has been subjected to a variety of experimental tests and found to yield a self-consistent quantitative picture of silver cluster formation in silver halides micro-

’

The Model

The Random Walk. Once the electron and hole have been placed in the conduction and valence bands by photon absorption, they begin to diffuse through the crystal. As they encounter impurities and phonons they are scattered, so that their movement can be modeled by a random walk. The mean jump distance in this random walk in each of the x, y, and z directions is a function of the diffusion coefficient, D,of the carrier and the time between jumps, t, as indicated in the following equation.

Overview of Mechanism of Silver Cluster Formation. Light absorption creates free electrons in the silver halide conduction band and free holes in its valence band. The indirect bandgap and momentum conservation rules prevent the direct recombination of the electron and hole, although they can recombine in other ways, as we will see below. The electron and hole begin to execute a random walk through the microcrystal. Electrons and holes can be trapped at lattice defect sites, such as kink sites at the surface. These kink sites are common features on ionic crystal surfaces and can be visualized as an incomplete step. They would be represented at the terminus of the incomplete step by either a silver ion (positive kink) or a bromide ion (negative kink), with a formal charge of 0.5 electrostatic units.6 These trapping sites usually provide only a small binding energy for the electron and hole, so that they quickly resume their random walk through the microcrystal via the conduction or valence bands.

The diffusion coefficient is related to the mobility of the carrier, which is in turn related to the interaction between the carrier and the surrounding lattice. As with most crystalline solids, the mobility of the electron is greater than that of the hole, so that the electron makes larger jumps than the hole. A result of this difference is that the electron traverses greater distances in a fixed time than the hole. This difference has important consequences for the efficiency of silver cluster formation, as we will see below. Trapping and Detrapping. The rate of trapping in crystalline solids depends on several factors. Foremost are the velocity with which the carrier is moving through the crystal, the cross section for carrier capture, and the density of the traps.I2 Carrier velocity can be determined from the Maxwell velocity distribu-

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4416 J. Phys. Chem., Vol. 99, No. 13, 1995

tion equation and is controlled by the mass of the carrier and the temperature. The cross section for carrier capture depends upon whether or not the trap has an excess charge relative to the surrounding lattice. If there is no charge, then the effective radius of the trap is just the lattice spacing. If there is an excess charge, then Coulomb factors become important. In this case the radius of the trap can be found by assuming capture of the carrier occurs when the Coulomb attraction just balances the thermal energy available, kT, where k is Boltzmann’s constant and T is the temperature. The resulting equation is

r = qtqce2/kTc where the q’s are the units of electrostatic charge e on the trap, qt, and the carrier, qc, and E is the dielectric constant of the crystal. Equation 2 is important in the case of ionic crystals, such as silver halides, in which lattice defects often provide the most effective traps and usually have an uncompensated charge. Release of a carrier from a trap is also important in the model where initial stages are reversible. The time a carrier spends in a trap, z, is determined by the trap depth, AE, the temperature, and the attempt-to-escape frequency, v, using the following equation z = v-‘ exp(AElkT)

(3)

The attempt-to-escape frequency represents the number of times per second that the carrier tries to escape from the trap. Because thermal energy is involved in this process, the frequency is related to the vibrational frequency of the surrounding lattice ions. The trap depth represents the energy difference between the trapping level and the bottom of the conduction band, in the case of electron traps, or between the trapping level and the top of the valence band in the case of hole traps. In those cases where the escape is from a charged trap, the trap depth also includes the thermal energy to overcome the Coulomb attraction between the carrier and the trap. For carrier trapping at charged lattice imperfections, the binding energy (or trap depth) of the carrier AE is readily calculated using a hydrogenic approximation.I2 That is, the binding energy is that of the electron in a hydrogen atom, 13.6 eV, but modified by the charge on the trap, if different from unit charge, the dielectric constant, and the ratio of the effective mass mef$3 to the free mass m of the carrier. The equation used for this calculation is then

meffqtqc (13.6 eV) me 2

AE=-

(4)

For values typical of silver halide microcrystals, AE is on the order of a few hundredths of an electronvolt for electrons but somewhat higher for holes because of their larger effective mas2 Silver halide microcrystals as grown, but without any further treatment, are very inefficient at forming silver clusters; a great majority of the electrons and holes recombine. Efficient silver cluster formation can be achieved by “chemically sensitizing” the microcrystals. In this process a chemical reaction between the microcrystal and a suitable reagent leads to the formation of inorganic clusters on the surface of the micro~rysta1.l~ Because typical reagents contain labile sulfur and/or gold, the nominal cluster building block is Ag2S or AgAuS. There probably is a size distribution of these clusters, but the details are unknown. Chemical sensitization creates electron traps that

are deeper than those traps caused by lattice defects,I5 and this increases the efficiency of silver cluster formation.I6 Atom Formation and Decay. As with trapping, atom formation is driven by electrostatic forces. Beginning with an empty trap having a formal charge OSe, the trapping of an electron brings the charge to -0.5e. This charge state now attracts a mobile interstitial silver ion to give a charge state of 0.5e. This alternating charge character that drives both electron trapping and atom formation is an essential element of the nucleation-and-growth model. A detailed picture of atom formation is unavailable. An interstitial silver ion, because it is not a lattice ion, has its empty 5s orbital lying below the conduction band and quite possibly below the energy level associated with the trapped electron. Once the Coulomb force has brought the silver ion into the region of the trapped electron, there would be a tendency for the electron to move to the 5s orbital of the silver ion to form a silver atom. The end result would be a silver atom adsorbed to the microcrystal surface adjacent to a positive kink site. Referring to this state as a silver atom may be an oversimplication. Perhaps a more correct view would be to visualize the state as an electron shared between a lattice defect and a silver ion, this species being held together by Coulomb forces. This may explain why it has been so difficult to detect the intrinsically paramagnetic silver atom by electron spin resonance measurement^,'^ although its inherent instability would surely be a contributing factor. In early considerations of the efficiency of the photographic process in silver halides, it was recognized that the photoinduced formation of silver clusters proceeded with a very efficient concentration process, even though the quantum yield of silver atoms might be 10w.I~That is, in most situations the result of exposure was one or a few silver clusters, as revealed by electron microscopic studies.I9 This could only happen if the silver atom was inherently unstable, allowing a rather efficient concentration process for the photoproduct. Likewise, it was recognized that the photographic process was extremely inefficient at very low irradiance.20 This too can best be explained by assuming the silver atom is unstable and that recombination represents a pathway for loss of the released electron.20x21 The decay mechanism of the silver atom probably is just the reverse of the formation process. An equation similar to eq 3, with AE replaced by the binding energy of a silver ion to the trapped electron, could be used to calculate the lifetime of the silver atom. Thermal fluctuations cause the silver ion to overcome the binding energy and to migrate away from the site of the trapped electron. Unfortunately, measurements of this binding energy are not available. Quantum mechanical calculations based on the CNDO method, however, do suggest that, at least on a relative scale, the silver ion has a considerably lower binding energy in the atom state than it does in silver clusters.22 Nucleation and Growth. The nucleation stage of silver cluster formation involves electron capture at the “deep” trapping level represented by single silver atoms. The difference between deep and “shallow” trapping is determined by the amount of energy that must be dissipated by the electron as it moves from the free band to the trapping leve1.10~23~24 If the required energy loss involves less than 0.05 eV, then it is shallow trapping, and the electron can easily transfer this energy to the lattice as it makes the transition to the trapping level. Deep trapping occurs when greater energies must be dissipated and the electron cannot do so while making a single transition. Instead, a multistep transition must occur. This is the case for electron trapping at a silver atom where, despite the instability of the silver atom

-

Ag Cluster Formation on AgBr Microcrystals

J. Phys. Chem., Vol. 99, No. 13, 1995 4417

I Conduction Band

Conduction Band

!V e

e

Shallow Trap

w

Relaxed State Silver Atom Level I

I

Configuration Coordinate

+

Figure 1. Configuration coordinate diagram for an electron trapped at a charged lattice defect.

ConfigurationCoordinate

+

Figure 3. As in Figure 2, but showing a much deeper relaxed state of the trapped electron due to chemical pretreatment of the microcrystal by sensitizing reagents.

t

*

P

e

w

U

e

r Atom Level

ConfigurationCoordinate

+

Figure 2. As in Figure 1, but also including the potential energy surface for the silver atom state. The crossing point C indicates the energy at which the electron may transfer from the relaxed trapped state to the silver atom state.

mentioned above, energy dissipation of much more than 0.05 eV is required.I0 Analogous to electron transfer between atoms, ions, or molecules, where potential energy surfaces are useful in visualizing the controlling factors, electronic transitions involved in electron trapping are best seen as transitions between potential energy surfaces. Initial trapping at lattice defects involves a Coulomb attraction between the electron and the charged defect. But, the electron in the region of the lattice defect introduces a distortion in the lattice as the ions now rearrange to minimize the energy. This provides a deeper trap for the electron by a lattice relaxation process as indicated in Figure 1.Io A thermal barrier to lattice relaxation is also included in Figure 1, where the configuration coordinate is a one-dimensional representation of a multidimensionalrearrangement of lattice ions. This picture of lattice relaxation is a general one for crystalline solids, but the details as they apply to electron capture in silver halides are ~ p e c u l a t i v e . ~ ~ Formation of a silver atom introduces an additional potential energy surface, as shown in Figure 2.1° The trapped electron in the relaxed state now has two choices. It can go back over the thermal barrier to the unrelaxed state, and then to the conduction band, or it can cross over to the silver atom potential energy surface at the crossing point C , and then along the silver atom potential energy curve to the minimum-energy point by losing energy to the lattice in stepwise fashion. As sketched in Figure 2, this latter process is of low probability, and in a large majority of situations the trapped electron will move back to the unrelaxed state, and from there to the conduction band. Hence, the nucleation stage of silver cluster formation is assumed to be inefficient.'O It is precisely this bottleneck that chemical sensitization is designed to overcome. The improved efficiency of the nu-

cleation stage that occurs upon chemical sensitization is attributed to the increased trap depth that the silver sulfide or silver gold sulfide clusters provide, and the resultant effect on the configuration coordinate diagram of the trapped electron as shown in Figure 3.1° The crossing point C to the silver atom curve is now in a more favorable position, and the electron can more easily move from one surface to the other. In contrast, molecular orbital calculations on silver clusters having two or more atoms10.22suggest that there are higherlying vacant orbitals that probably intersect the potential energy surface of the trapped electron at a more favorable energy position so as to facilitate electron transfer to the silver cluster potential energy surface. Thus, the growth steps of silver cluster formation are assumed to proceed with high electron-trapping efficiency. It is the nucleation stage that represents the bottleneck in silver cluster formation. Earlier simple semiclassical treatments of silver clusters adsorbed to AgBr microcrystals have suggested that the electronaccepting level of the clusters monotonically moves deeper into the AgBr bandgap as the cluster size increases and that this level asymptotically approaches that for bulk silver.26 More recently, simple molecular orbital calculations on isolated silver clusters have shown a distinctive odd-even alternation in the electron-accepting and this has been confirmed by more sophisticated calculations on silver clusters adsorbed to AgBr surfaces.22 These calculations indicate that the electronaccepting level of the odd size cluster will lie lower than the even size cluster. This picture has been confirmed by gas-phase studies of silver clusters,28although attempts to confirm it for silver clusters adsorbed to AgBr microcrystals have been inconclu~ive?~For the model discussed here we simply assume that all clusters have an electron-accepting level deep enough within the bandgap to qualify the clusters as permanent electron traps. Recombination. The two possible pathways for recombination both involve the trapping of one carrier, followed by the capture of the complementary free carrier. A predominant trap for holes is the negative kink site at the surface. Holes have a higher effective mass relative to electrons, which, according to eq 4,will lead to a deeper trap for holes at negative kinks than electrons at positive kinks. For microcrystals without chemical sensitization, the more deeply trapped carrier will then be the hole, so the predominant pathway for recombination will be free electrodtrapped hole. Upon chemical sensitization, the electron spends increasing amounts of time in traps as the density and/or depth of electron traps increases. Eventually, the predominant pathway shifts to free holekrapped electron. Optimum chemical sensitization is achieved by causing electrons

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4418 J. Phys. Chem., Vol. 99, No. 13, 1995 to spend enough time in traps so as to minimize the first pathway, but without increasing the probability of the second pathway. The two-step trapping-capture process of recombination leads to the formation of an exciton state. The decay of the exciton is nonradiative at room temperature, although some luminescence can be observed at liquid nitrogen temperatures. The details of the nonradiative pathway are obscure, but the dissipation of energy appears to be efficient. The recombination process is assumed to have a high efficiency for the electron transition to the deep level, similar to that for the growth step.I0

WHILE step Increment simulation time

If time for a photon absorption Absorb photon

If time jump possible

Computer Simulation A Monte Carlo simulation of the type required for silver cluster formation in AgBr microcrystals can be very computationally intensive. Many years ago a simulation code was written for a mainframe computer that took into account all the processes described above! It was quickly discovered that it would require impossibly long run times. The time scales for the reversible steps are much faster than those for the irreversible steps, so that the former consumed all the available run time. As a way around this barrier, a steady-state assumption was applied to the reversible steps, and the simulation focused on the irreversible stepsS4This has allowed many valuable studies to be carried out over the ensuing year^."^^^ Despite this success, some limitations still remain with this approach. Primary among these is the inability to explicitly study the effect of the size and shape of the microcrystal on the efficiency of silver cluster formation. Additional deficiencies are the study of trap depth and density and silver atom formation and decay in only an indirect way. These limitations prevent the use of a truly computer-aided design tool in developing improved microcrystals for photographic applications. The simulation approach discussed in this paper was designed to circumvent these limitations so as to allow a very detailed study of the parameters controlling silver cluster formation. An essential feature of the Monte Carlo process as applied to simulating the reponse of film layers is the repeated simulations over an ensemble of microcrystals. This obviously increases the run time by a factor roughly equal to the number of microcrystals in the ensemble. Typical ensemble concentrations are 400 and preferably 1000 to achieve any reasonable level of statistical reliability for comparison with experimental data. A way around this obstacle is to realize that all the microcrystals in the ensemble act independently of each other during exposure. With today’s computing technology, we can assign different processors in a parallel processor to simulate the silver cluster formation in individual microcrystals. In this way we can achieve a significant speedup of the simulations that is in proportion to the number of processors available. Hardware. Our parallel processor has been described in detail el~ewhere.~’ It is composed of 64 T-8OO3* transputers placed on four boards which are in turn connected to another board containing the master transputer which oversees the work being done by the transputer array. The card cage containing these transputer boards is connected via a VME bus to a Sun 3 workstation which acts as the user interface. Benchmarking experiments have shown that similar simulations run on an IBM 3090 mainframe require about 17 times more run time than when run on our parallel processor. Software. The transputers are programmed in OCCAM, a language especially designed for programming transputer arr a y ~ This . ~ ~is no longer a requirement, as other more common high-level languages can now be used. However, to harness the full power of parallelism achievable with transputers,

Do time jump Electron loop (Fig.5 ) Hole loop Figure 4. OCCAM pseudocode describing the main loop of the simulation code. OCCAM should be the language of choice. The user interface on the Sun workstation is written in C and uses a special library to allow for commands dealing with data transmission from and to the transputer array. The simulation program consists of over 2500 lines of OCCAM code. The flow of the program essential for the discussion in this paper is shown in Figures 4-8. The indentation of the text in these figures is characteristic of the OCCAM programming environment. All lines of code having a similar indentation belong to the same process and are initiated by the first preceding outdented line. This feature is mandatory in OCCAM and leads to more readable program structures. The core of the program involves the successive incrementing of the time step used to simulate time evolution (Figure 4). The WHILE statement is a OCCAM conditional process which continues until the Boolean variable “step” is no longer true. Within this loop photon absorption, time jumping, and surveying the status of electrons and holes take place. The IF statement is another OCCAM conditional process which leads to the execution of the following indented process(es) only if the IF statement is true. The internal simulation time is incremented by the chosen time step every time the status of the electrons and holes is surveyed. In those cases where there are no free carriers, the simulation code determines the next instance when a carrier will be freed, an electron will convert to a silver atom, an atom will decay, etc., and moves the simulation time to correspond with the next action to occur. Employing this time-jumping strategy greatly decreases the run time needed for our simulations. The simulation code has two main loops-one for the electron (Figure 5) and a similar one for the hole (not shown in flow diagrams). After every time step each loop is executed for each of the carriers that have been created. The SEQ statement is an OCCAM replicator, and it acts much like a DO loop in FORTRAN, except that the program execution cannot be made to jump out of the loop before its scheduled termination. The loop must be completely executed each time it is initiated. For each carrier that has been created by photon absorption, the code determines its status. For the electron these are (1) no longer active (because took part in nucleation, growth, or recombination), (2) atom state, (3) trapped state, and (4) free

J. Phys. Chem., Vol. 99, No. 13, 1995 4419

Ag Cluster Formation on AgBr Microcrystals IF electron free

SEQ ipart = 1 FOR npart

IF electron DO longer active (status 1)

Check for closeness to surface

Add to inactive caunter

IF within trap radius of surface

IF electron in atom state (status 2) Check for trapping by silver center

IF simulation time > atom decay time Check for trapping by empty trap Atan to trap transition

IF trapped IF electron in trap state (status 3) Change coordinates to that of trap or silver center

IF simulatioo time > escape time

IF first electron at trap

Electron escapes

Change electron status from “free” to “trapped”

IF simulation time > atom formetian time

Calculate neutralization time

Atom formation

Calculate escape time

IF electron free (status 4)

IF not first electron at trap

Electron makes a jump

WHILEfree

IF nucleation possible

D e t e r ” jump distance

Change status of electron already at trap

Che& for trapping (Fig. 6)

Increment number of electrons at trap

Che& for recombination (Fig. 7)

IF jump takes electron out of microcrystal

IF growth possible Che& for trapping

Increment number of electrons at trap

IF no trapping Bring electron back inside by reflececting trajectory off surface

Figure 5. OCCAM pseudocode describing the loop that checks the status of the electron and performs the jump in position if the electron is free (bottom section).

Figure 6. OCCAM pseudocode describing the electron trapping algorithm indicated at the bottom of Figure 5 . If trapping occurs, then the program must determine what kind of trapping: empty trap, nucleation, or growth event. IF electron free

state. The status possibilities for the hole are the same, except for the atom state. When in state 2 or 3, the simulation checks to see whether it is time for a given event to occur (Figure 5 ) . For example, when in state 3, is it time for release of the electron to the conduction band or formation of a silver atom? If the carrier is free (state 4),then the simulation code executes the jump section, moving the carrier to a new position. After the jump, the code checks for any carrier trapping (Figure 6) or recombination (Figure 7). Classical Approximation. The simulation treats the transport of electrons and holes in a classical way. The neglect of quantum confinement effects can be justified by considering the extent of the electron wave function relative to the size of the microcrystal. The extent of the wave function can be estimated from its wavelength using the de Broglie equation

IF recombination radius > 0.0 Determine distance between electon and trapped hole

IF distance > recombinationradius Electron not in recombinationsphere Determine orthogonal distance between tajectory and recombinationcenter

IF distance < recombinationradius Recombination occurs

IF &lance > recombinationradius continue

(5)

Figure 7. OCCAM pseudocode describing the recombination algorithm indicated at the bottom of Figure 5 .

where 1 is the wave function wavelength, h is Heisenberg’s constant, and v is the electron velocity derived from the Maxwell velocity distribution, using the effective mass of the electron. For values appropriate to AgBr, eq 5 yields a wavelength of 0.02 pm, Since the lower limit of microcrystal diameter of interest in our simulations is 0.1 pm, we can ignore quantum confinement effects.

Photon Absorption Sequence. When averaged over an ensemble of microcrystals, the absorbed photon distribution should obey the Poisson distribution. To achieve this goal in the simulations, a photon absorption sequence is calculated for each microcrystal before the simulation of silver cluster formation is commenced. Two rates are defined. The first, rl, is just the mean number of photons as specified in the input

il= h/m,,v

4420 J. Phys. Chem., Vol. 99, No. 13, 1995

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A A 100

'

,

A

*g,

, I . . . '

0.10

0.01

1.oo

Jump Distance, km

Figure 8. Efficiency of silver cluster formation vs the electron jump distance. Labels indicate the silver cluster size. Symbol size represents the statistic uncertainty in the simulated quantum sensitivities.

parameter list divided by the exposure time. This is effectively the irradiance for a particular microcrystal. The second rate, 7-2, is that for the null event and is made 10 times larger than 7-1. A uniform random number generator is used to produce a random number, X I ,between 0 and 1. This random number is then scaled by multiplying it by the total rate, rt, which is the sum of 7-1 and r2. The following equations are used to decide whether a photon absorption has occurred.

x, 5

I,,

photon absorption

(6)

x, > r I , nullevent The time between photon absorptions, t , is calculated from the total rate and another random number, x2, from the following equation4 t = -In(;) 1 rt

(7)

Thus, the time between photon absorptions follows an exponential distribution, as would be expected for events that occur in a random sequence. The photon absorption algorithm is placed within a WHILE process and continues to generate photon absorption events until the sum of the time between absorptions equals or exceeds the exposure time. This time sequence is then stored in an array. After every time step in the simulation, the program refers to this array to determine whether it is time for another photon absorption. Carrier Diffusion. Following a photon absorption event, an electron and hole are added to the counters keeping track of the total number of electrons and holes. Their coordinates are assigned by randomly choosing a location within the microcrystal. Currently, the absorption is considered to be uniform, although options exist within the program for nonuniform absorption. The mean jump distance in each of the x, y , and z directions is determined by eq 1, in which the diffusion coefficient and time step are input parameters. The actual jump distance is normally distributed around this mean value by multiplying it by random numbers normally distributed around l.0.34 The direction of the jump is also randomized by making the jump distance a negative quantity if a uniformly distributed random number is < O S and leaving it positive if it is 20.5. Trapping and Detrapping. In the simplest case carrier trapping occurs only at the surface of the microcrystal. Traps are characterized by a trap radius and a trap density, both input parameters, which determine the trapping probability. When

the carrier position is within a trap radius of the surface, trapping is possible (Figure 6). Likewise, if the carrier trajectory on its latest jump passes outside the microcrystal, trapping is also possible (Figure 5 , bottom section). In either case the trapping probability determines whether or not trapping will occur. If traps are also present in the interior of the microcrystal, then the situation can become much more tedious and timeconsuming. This is because the possibility of trapping must be checked after every jump of the carrier. As a way around this difficulty, we have used a system of concentric shells spanning the volume of the microcrystal to simulate internal trapping by randomly placed traps.35 As a carrier trajectory crosses a particular shell, the carrier has a probability of being trapped at that shell which is determined again by the density and the radius of traps. This scheme could easily be extended to a situation in which the traps are spatially inhomogeneous. Electron trapping at internal locations can lead to formation of internal silver clusters. Evidently, lattice ions are pushed out of their normal positions to make room for the growing silver cluster.36 In general, the formation of internal clusters is undesirable because they are undetectable by conventional photographic developers. Specially formulated developers can be used to detect these internal clu~ters,~' but such developers are impractical for commercial purposes. Upon trapping, whether it be at the surface or at one of the concentric shells mimicking internal traps, a trap residence time must be determined (Figure 6). The mean residence time is determined by eq 3, where the trap depth is an input parameter and the attempt-to-escape frequency is an internally fixed value of 10" s-l. The actual residence time is selected from an exponential distribution in which the residence time from eq 3 is the mean of the distribution. The computed residence time is added to the simulation time to determine at what point in the future the carrier may escape from the trap and resume its random walk. The above value for the attempt-to-escape frequency is about a factor of 10 smaller than the maximum phonon frequency, which is sometimes used for this parameter. However, smaller values than either the maximum phonon frequency or our chosen value have been observed by others in various silver halide systems.38 Likewise, we are concerned with transitions occurring at surface states in contact with gelatidwater, which could play a role in lowering the frequency. A further complication is that this parameter could be a function of the specific properties of the trapping site and, thus, may vary within a given system. Although experimental determinations of this parameter have been made for certain traps,38 values are generally unavailable for centers created by chemical sensitization. Thus, the appropriate attempt-to-escape frequency is quite uncertain, and our chosen value merely represents a compromise among the many possible values. Atom Formation and Decay. While the electron is in a trap, it is possible for a silver atom to form by the combination of the trapped electron and an interstitial silver ion that has migrated to the site of the trapped electron. This event is usually referred to as the neutralization of the trapped electron, and a neutralization time must also be calculated upon electron trapping (Figure 6). The mean time for this neutralization is an input parameter. As with trap residence times, the actual time for the neutralization step is exponentially distributed about the mean neutralization time. Both the trap residence time and the neutralization time are calculated upon trapping of an electron. At this point the shortterm fate of the electron is determined. If the trap residence

J. Phys. Chem., Vol. 99, No. 13, 1995 4421

Ag Cluster Formation on AgBr Microcrystals time is less than the neutralization time, the electron will escape to the conduction band. Otherwise, a silver atom will form. Once a silver atom is formed, its lifetime must be calculated. Once again, the actual lifetime is exponentially distributed around a mean lifetime which is an input parameter. When the silver atom decays, the silver ion migrates away from the site, leaving behind a trapped electron. The trap residence time of this electron and the neutralization time are again calculated as above. Thus, the silver atom may re-form, depending on the results of these calculations. Nucleation. This event involves the electron trapping at a silver atom site and is part of the electron trapping process that is included in the trap checking algorithm (Figure 6 ) . The trapping radius can be taken as the same as that for trapping at an empty trap. This assumes that the trapping efficiency is unity and that there are no barriers to the electron making the transition from the shallow to the deep trapping level of the silver atom. If this transition is not a fully efficient process, then the trap radius can be made smaller to reflect this difference between trapping at a silver atom site and trapping at an empty trap. At best, the nucleation event must compete with all the other possible trapping events, indicating nucleation happens only rarely. Clearly, the rate of nucleation will depend upon the number of silver atoms and their lifetime, as well as the effective trap radius. Because the electron trapped at a silver atom is considered to be in an irreversible state, the migration of the interstitial silver ion to form Agz is not explicitly followed in the simulation. Rather, an Agz is assumed to be formed immediately after the electron trapping event, and this cluster is considered to be permanently stable. Thus, at this point the final fate of both electrons-the first one leading to the silver atom and the second one leading to an Agz-is known. Growth. Electron trapping at Ag2 and larger clusters constitutes a growth event and is also part of the electron trapping process that is included in the trap checking algorithm (Figure 6). Because the silver cluster formation is assumed to occur at charged trapping sites, where the effective capture radius of the trap is much larger than the lattice dimensions of the site, the trap radius of a growing silver cluster is assumed to be the same as that for the trapping site which initiated silver cluster formation. In accordance with the above discussion, the efficiency of the electron transition to the deep level of the growing silver cluster is assumed to be unity. Likewise, interstitial silver ion migration is not explicitly followed. The limiting factor on the number of growth events is the availability of electrons in the conduction band. Once all the electrons are consumed by nucleation, growth, or recombination, the simulation code surveys the different counters keeping track of silver clusters. The number of silver clusters and the number of atoms per cluster are used to classify the cluster distribution for the microcrystal. These data are combined with similar data for other microcrystals in the ensemble to determine the cluster distribution. Recombination. When a carrier approaches the surface of the microcrystal, besides checking for carrier trapping, the simulation code also checks for an oppositely charged carrier already trapped at the surface. If such is found, then there exists a possibility for recombination between the two carriers. The radius of the sphere surrounding the trapped carrier is determined by the recombination radius, an input parameter. The checking for recombination has two parts (Figure 7). If the electron is within the recombination sphere of a trapped hole, then recombination occurs with unit probability. If the electron’s trajectory took it through the recombination sphere,

TABLE 1: Information Returned to Master Processor from Individual Processors information

use

cluster distribution number of photons number of recombinations number of nucleations total distance traveled

determines microcrystal sensitivity determines microcrystal sensitivity extent of recombination tendency toward multiple nucleations how well does electron sample the entire microcrystal availability of electron for motion time scale for silver cluster formation

number of jumps simulation time

recombination also occurs with unit probability. In the latter case the principles of solid geometry are used to calculate the orthogonal distance between the trajectory and the center of the recombination sphere. If this distance is less than the recombination radius, then recombination occurs. The same algorithm is used for recombination between a free hole and a trapped electron. In some cases a trapped carrier and an empty trap may be located in close proximity to each other so that trapping and recombination of the free carrier are both possible. To avoid any biasing, the simulation code has two pathways it may follow. In one pathway it checks for recombination and then trapping; in the other pathway the order is just the reverse. The selection of which pathway to take is made by generating a random number on the interval 0-1. If the number is 50.5, one pathway is taken. Otherwise, the other pathway is taken. Finishing Up. The simulation for a given microcrystal is concluded when there are no further reactions possible for the electrons, even though some holes may remain active. That is, the electrons have participated in nucleation or growth events or have recombined with holes. All other events are reversible, and if any are possible the simulation must continue. As each processor completes its simulation of the response of a microcrystal, it sends the results to the master processor that oversees the transputer array. The information returned to the master processor is given in Table 1. If more simulations are required, then the master processor sends instructions to the now-idle processor that it should commence another simulation. In general, the same parameters are wed, but a different initial seed for the random number generator causes a different sequence of events to occur. Once all the specified simulation runs have been completed, the master processor compiles the information from the individual transputers and sends the information to the host workstation for user access. This information is basically the same as for individual processors, but now averaged over all the microcrystals in the simulation ensemble. In addition, the cluster distribution, mean and variance are determined for the ensemble for each cluster size up to six atoms. At this point the simulation for one photon value is completed. Typically, the simulations commence with the mean number of photons set at two. Upon completion at the first setting, the simulation program will increase the mean number of photons to four and by continuing factors of two thereafter. After each increase in the photon value, the simulation is restarted for the specified number of microcrystals. The end of the simulation is determined when a specified fraction of microcrystals having a certain size cluster or larger is reached. This fraction is an adjustable parameter and can vary widely, depending on what the simulation is intended for.

Parameters The parameters that control the simulation, along with some typical values and their source, are listed in Table 2. In this section I will discuss some of these parameters in more detail.

4422 J. Phys. Chem., Vol. 99, No. 13, 1995

Hailstone

TABLE 2: Parameter Values Used in the Simulations parameter value comments 10-12-10-11 time step adjustable diffusion coefficient, los- io9 pm2/s experimental mobility, electron calculations diffusion coefficient, hole 106-107 Um2/s experimental mobility, calculations trap depth 0.05-0.5 eV experimental data, calculations exposure time 10-6-103 s adjustable recombination radius 1.O- 10.0 nm charge dependent trap radius 1.0-10.0 nm charge dependent ionic neutralization time 10-4-10-7 s ionic conductivity atom lifetime 10-4- 100s very uncertain hole trap lifetime 10-9-10-7 s calculated trap density 10-6400 pm-2 adjustable adjustable number of microcrystals 400- 1000 0.1- 1.0bm adjustable size (cube edge length)

Time Step. This is used in eq 1 to calculate the jump distance in each of the x , y, and z directions. Ideally, the time step should be the collision or scattering time of the electron. This can be calculated in a classical approximation from the mean free path of the electron and its thermal velocity. This would lead to a value of about s. Unfortunately, such a short time would lead to very small jump distances via eq 1 and consequently very long run times. Since in many cases we are only interested in events that occur when the carrier reaches the surface of the microcrystal, a longer time step is useful, providing it does not introduce artifacts. An upper limit on the time step is the shortest expected time for any of the events occurring after carrier trapping. This turns out to be the trap residence time. If the time step, which determines how often the release from traps is checked, is comparable to or longer than the trap residence time, we will introduce artifacts. Using eq 3 and a trap depth of 0.05 eV, a mean trap residence time of 74 ps is found. Given that there is an exponential distribution around this mean time, if we take lo-" s as the upper limit of the time step, we should avoid artifacts from having a time step that is longer than the shortest expected trap residence time. Diffusion Constant. In principle, this parameter can be calculated from the experimentally measured mobility via the Einstein expression

D = ,u(kT/e)

(8)

where p is the mobility in cm2/(V s) and e is the electrostatic unit of charge. This transformation is complicated by several factors. There are two possible mobility values. The microscopic mobility relates to the carrier mobility for traveling between trapping events and is therefore determined by phonon scattering at room temperature. The drift mobility relates to the time spent traveling through the conduction band, as well as time spent in traps. Ideally, we would use the microscopic mobility to calculate the diffusion coefficient, but in order to be consistent the time step would have to be the time between phonon scattering events. Because we have already chosen to use a time step longer than this scattering time, a mobility including time spent in shallow internal traps would be more appropriate. Thus, the drift mobility would be a better approximation. However, these measurements are difficult for microcrystals, as evidenced by several different reported values.39 Ultimately, the methodology we have adopted is to use the diffusion coefficient as an adjustable parameter. A value is selected such that the mean jump distance in three dimensions, r, calculated from the equation

( r ) = d6Dt

(9)

is equal to 20% of the edge length of the model space used for the simulations. The time step t is chosen as described in the previous section. By experience, we have found that this procedure leads to reasonable run times without introducing undesirable artifacts. When a series of different size microcrystals are being studied, both the diffusion coefficient and the time step are changed by the same factor to achieve the 20% of edge length criterion for the mean jump distance. Holes are much less mobile than electrons in silver halides.@ To simulate this effect, the diffusion coefficient for the hole is set at 10-2-10-4 that of the electron in AgBr simulations. The difference in electron and hole mob es can have profound effects on the recombination processes in microcrystals. Establishing a more quantitative ratio of mobilities is very important for more realistic simulations. Trap Depth. This parameter is one of the more important ones in determining the efficiency of silver cluster formation. Many of the applications of the simulation code are to microcrystals which have been chemically sensitized. A variety of experimental techniques have been applied to determine the trap depths in chemically sensitized systems.I5 Typical values range from 0.1 to 0.5 eV. Because these values appear in a Boltzmann factor (eq 3), small changes in trap depth can produce large changes in trap residence time. A further complication can also be present. Chemical sensitization involves the formation of (AgzS), or (AgAuS), clusters on the microcrystal surface. The bulk of evidence suggests that the formation of these clusters proceeds by an aggregation process of unknown me~hanism.~' There likely is a size distribution of these clusters, and different size clusters are likely to have different trap depths.'5d Thus, there most probably is a distribution of trap depths, but its details are unknown. Results discussed in this paper assume a single trap depth and are therefore a simplification of the actual picture. Despite this simplification, the trends can be very enlightening. Trapping and Recombination Radii. In the simplest case trapping or recombination involve carrier capture at a site bearing an uncompensated charge. Thus, eq 2 is used to calculate the trapping or recombination radii. Taking surface kink sites with a charge of 0.5e as an example, eq 2 yields a value of 2.2 nm. Since from a charge standpoint a positive kink site has the same charge as a hole trapped at a negative kink site, this calculated value applies to both electron or hole capture at an empty trap and to recombination between a free electron and a trapped hole or between a free hole and a trapped electron. In all these cases, aside from a sign difference, the capturing site has the same Coulombic charge. Other factors could come into play to change these calculated trap and recombination radii. For example, surface relaxation would change the interionic spacing and alter the excess charge on the kink site. Recombination involves capture at a shallow trapping level to form the exciton state, followed by nonradiative recombination. If the efficiency factor is less than unity for the shallow to deep transition, then the recombination radius will be correspondingly reduced, even though the charge on the site may suggest a large radius. It appears that this latter situation is not applicable to recombination in silver halides.42 A simplification that can be used to speed up run times is to recognize that it is not the absolute value of the trapping and recombination radii that is important in determining the efficiency of silver cluster formation, but rather the relative radii. Using 5.0 nm for both trapping and recombination radii leads to a dramatic improvement in run time, but with the same results

J. Phys. Chem., Vol. 99, No. 13, 1995 4423

Ag Cluster Formation on AgBr Microcrystals as far as cluster formation is concerned. This procedure has been used in the simulations to be discussed in this paper. The reason for the improved run time is that the number of trapping or recombination events per carrier collision with the surface increases, leading to higher probability of the irreverisible events and correspondingly less time spent following the carriers on their random walk through the microcrystal. Ionic Neutralization Time. Decay of photoconductivity of AgBr microcrystals in the appropriate time regime is controlled by interstitial silver ion migration to the site of the trapped electron. The rate constant for this process has been measured at 106-107 s-I, depending on microcrystal size.43 This rate is controlled by the ionic conductivity of the microcrystal, which is determined by the product of the concentration of interstitials and their mobility. Most often it is the concentration of interstitials that can be influenced by various intentionally added organic silver ion complexing compounds. Thus, these compounds can affect the efficiency of cluster formation. An alternative view recognizes that the migration of the interstitial silver ion is due to Coulombic forces between it and the trapped electron. Analysis of this process has shown that the mean time for the neutralization step corresponds roughly with the dielectric relaxation time of the microcrystaLU Dielectric loss measurements on microcrystalline film layers have yielded values in the 106-107 s-l range, also depending on microcrystal size.45 Thus, there is a good correlation between the ionic and electronic relaxation times. Silver Atom Decay Time. Silver atoms are well recognized as unstable species on AgBr microcrystal surfaces. Intermittency studies, in which the normal exposure is partitioned into many smaller exposures of lower intensity, have been used over the years to investigate decay processes in AgBr microcrystals.46 As the time duration between intermittent exposures increases, a decrease in density produced upon development is often seen, giving the appearance of a decay profile. Decay times anywhere from milliseconds to several seconds have been reported. In some cases these decay times have been assigned to that of the silver atom. Given the reversible nature of the initial stages of silver cluster formation, it is likely that the electron undergoes many atom formatiodatom decay cycles before its final fate is determined.' Therefore, these intermittency experiments most probably measure an upper limit to the decay time, and the mean time could be orders of magnitude smaller. As a consequence, the atom decay time is not known with any degree of certainty. For all of the simulation results reported here, we have used s as the atom lifetime. Hole Trap Residence Time. The traps for holes are assumed to be negative kink sites on the surface. The expected trap depth after lattice relaxation is estimated from photoconductivity simulations to be 0.15-0.25 eV.47 Using eq 3 with the same 10" s-I attempt-to-escape frequency as for the trapped electron leads to a trap residence time of about 10-9-10-7 SKI.The simulations discussed here use the former value.

Results Jump Distance. The jump distance is determined by the diffusion coefficient and the time step via eq 9. As noted in the previous section, these parameters are adjusted to give a mean jump distance that is 20% of the edge length. In this way computation time is reduced, but we must be sure that this procedure does not introduce artifacts into the simulation. To verify that this is the case, we ran simulations in which the jump distance was smaller than that calculated by our usual procedure. Shown in Figure 8 are efficiency results for a 0.5

TABLE 3: Comparison of Simulated and Theoretical Jump Distances jump distances, bum time step, s simulation theon, (ea 9) 10-1' 0.17 0.17 ~~

10-12 10-13

10-14

0.052 0.016 0.005 1

0.055

0.017 0.0055

pm cubic microcrystal as a function of jump distance. The efficiency of silver cluster formation is independent of jump distance between 0.17 and 0.0055 pm. The latter distance is within a factor of 2 of the mean free path of an electron in AgBr using a classical approximation, in which the mean free path is determined by phonon scattering events. According to the results in Figure 8, we can choose a larger jump distance than the mean free path and achieve the same efficiency of silver cluster formation, but with a significant decrease in run time. The simulations used for the data in Figure 8 were also used to verify that the program was computing jump distances whose mean was consistent with eq 9. The program provides the total distance traveled by a carrier and the total number of jumps made. Simple division provides the mean jump distance. Table 3 summarizes the results for the four cases studied, with the mean jump distance from the simulations being an average over about 800 electrons. The agreement between simulation and theory (eq 9) is excellent and shows that the program is calculating the parameters for the random walk in the correct way. Characteristic Curves. To illustrate the effect of the density and depth of electron traps, we show the characteristic curves for a high irradiance condition in Figure 9 for varying trap density and Figure 10 for varying trap depth. Parameter values are given in Table 4. In these figures the efficiency increases as the curve moves to the left, where fewer photons are required to produce the silver clusters. The efficiency of forming twoatom clusters (Figure 9A) shows the smallest and that for forming five-atom clusters (Figure 9D) shows the largest dependence on trap density. The effect of trap density is explained by recognizing that silver clusters form by a nucleation-and-growth sequence. This sequence involves both electronic and ionic events, the latter occurring on the microsecond time scale. Under high irradiance, all the electrons and holes are generated before any nucleation or growth events can occur. Because we assume in these simulations that the efficiency of electron capture at a silver atom center is high, the nucleation will compete very favorably with growth. In effect, the growth stage is inefficient because the large concentration of electrons favors nucleation over growth. In Figure 9A we find very little dependence of the efficiency of Ag2 formation on trap density. This is because there are no required growth steps, and so the number of nucleation sites plays a minor role. The small dependence observed relates to minimization of the recombination efficiency. Higher trap densities mean that the electrons spend more time in traps, with a corresponding reduction in free electrodtrapped hole recombination. However, there is an upper limit to trap density (1600 pm-*), beyond which free holekrapped electron recombination becomes important. On the other hand, Figure 9D shows that the highest efficiency is obtained with the lowest trap density. Limiting the number of potential nucleation sites makes growth to a fiveatom size more efficient. This result appears to conflict with that for Agz where 1600pm-2 gave maximum efficiency. But, the contribution to the overall efficiency by efficient growth

Hailstone

4424 J. Phys. Chem., Vol. 99, No. 13, I995 1.o

0.8

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for this high-irradiance condition is much more important than any reduction in recombination losses at the nucleation stage by a high trap density, and so minimizing the number of potential nucleation sites is more important. Figure 9B,C shows intermediate behavior because the number of required growth steps is smaller than three in these cases. Figure 10 pertains to a fixed trap density of 400 ,um-2. The two-atom clusters show maximum formation efficiency at intermediate trap depths of 0.2-0.3 eV (Figure 10A). At 0.1

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Figure 10. As in Figure 9, but labels now indicate the trap depth. Trap density is 400/pm2.

eV trap depth the electron spends too little time in traps making atom formation, and therefore nucleation, less efficient. The same explanation cannot be used for the 0.4 eV trap depth, where atom formation is expected to be the most efficient. Rather, the inefficiency results from an enhancement of free holehapped electron recombination because the electrons spend too much time in the trap state. Similar trends were observed for the three-atom clusters. As the cluster size increases, and therefore the number of required growth steps increases, the trap depth that leads to optimum efficiency decreases to 0.1 eV (Figure 10C,D).

Ag Cluster Formation on AgBr Microcrystals

J. Phys. Chem., Vol. 99, No. 13, 1995 4425

TABLE 4: Parameter Values Used in the Simulations of Figures 9-12 parameter time step diffusion coefficient, electron exposure time

value lo-" s 4 x 1O8pm2/s 4 x 106pm2/s 0.1-0.4 eV 10-6 s

recombination radius trap radius

5.0 nm 5.0 nm

diffusion coefficient, hole trap depth

Although this small trap depth leads to poor efficiency for the smaller clusters because of inefficient silver atom formation, it also leads to a situation in which growth is more competitive with nucleation. As we saw in Figure 9D, this competition plays a key role in determining the efficiency of cluster formation at high irradiance. A general trend observed in Figures 9 and 10 is that larger clusters form with less efficiency than smaller clusters. That is, for a fixed density and depth of traps, the curves corresponding to different cluster sizes are displaced along the x axis by considerably more than the one photon increment that might be expected between curves corresponding to one-atom difference. The explanation for this behavior comes from the effect of recombination. As nucleation and growth proceed, the number of holes available for recombination increases because there is one excess hole for every atom in a silver cluster. This makes the formation of each successive cluster size less efficient than the prior one. Cluster Distributions. The characteristic curves just discussed do not say anything about the number of clusters per microcrystal but merely relate to that fraction of microcrystals which have at least one of the requisite clusters for catalyzing development. For photographic development of microcrystals this is sufficient. But, for mechanistic understanding it is useful to look at the cluster distributions. The simulation code is designed to provide this information. Figure 11 shows the mean number of clusters as a function of the mean exposure of the microcrystal for different trap densities corresponding to those discussed in Figure 9. The slope in this type of plot is in units of clusters per photon, so that the reciprocal of the slope is in units of photons per cluster. The latter relates to the efficiency of cluster formation. The smaller the reciprocal slope, the fewer photons are required to create a given cluster, so that the efficiency is higher. Therefore, the higher the slope on these plots, the higher the efficiency of cluster formation. Two-atom clusters (Figure 11A) show a maximum efficiency at 400 traps pm-*. Higher trap densities lead to increased recombination via the free holeltrapped electron pathway. A lower trap density increases recombination via the free electron/ trapped hole pathway. For both three- and four-atom clusters (Figure 11, B and C) the efficiency of cluster formation decreases monotonically as the trap density increases. This behavior is related to the characteristic curve behavior discussed in the previous section. One contributing factor is that higher trap densities provide more sites for nucleation and lead to inefficient growth. This is why the slopes are lower for the four-atom clusters than for the three-atom ones. Another contributing factor is that higher trap densities cause the electron to spend more time in traps, enhancing the free holehrapped electron recombination pathway. Figure 12 is analogous to Figure 11, except that now the effect of trap depth is being explored. In general, all three cluster sizes show a maximum efficiency at 0.2 eV, although 0.1 eV at small numbers of photons per microcrystal is best for Ag4 (which is consistent with Figure 10). The smallest trap depth

parameter ionic neutralization time atom lifetime hole trap lifetime trap density number of microcrystals size (cube edge length)

value 10-6 s 10-4 10-9 s

100-6400 pm2 400 0.5 pm

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of 0.1 eV leads to poor nucleation efficiency, whereas the higher trap depths of 0.3 and 0.4 eV enhance the free holeltrapped electron recombination. For the three- and four-atom clusters (Figure 12, B and C), the larger trap depths also promote additional nucleation events which decreases the growth efficiency. This effect is not important for the two-atom clusters where only nucleation is involved. Efficiency. In Figure 13 we show the dependence of the quantum sensitivity (QS)on trap depth and trap density for three-atom clusters. The optimum number of traps for maximum efficiency depends on the chosen trap depth. Shallow traps (0.1 eV) require a large trap density (1000-3000 pm-2), whereas deep traps (0.3 eV) require a low trap density (100-

Hailstone

4426 J. Phys. Chem., Vol. 99, No. 13, 1995

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Figure 13. Efficiency of Ag3 formation plot showing the dependence on trap depth and trap density for a 0.5 pm cube.

500 ,um-2). The ascending portion of these curves shows increasing efficiency due to increasing efficiency of nucleation of silver clusters resulting from decreasing free electrodtrapped hole recombination. After going through a maximum, the descending portion of the curves shows decreasing efficiency

due to increased free holeltrapped electron recombination. The deepest trap studied, 0.4 eV, would presumably require only a few traps per microcrystal to achieve high efficiency. Even at optimum trap depth and trap density the best quantum sensitivity is about eight photons per microcrystal (0.2 and 0.3 eV trap depths). At this optimum position the free electrod trapped hole recombination pathway has been minimized, while keeping the free holeltrapped electron recombination pathway from becoming important. Even under these optimized conditions, there is still some residual recombination occurring since a fully efficient condition would have an efficiency of three photons per microcrystal. Thus, it appears that by simply adjusting trap depth and trap density, one cannot achieve a fully efficient situation in which all the photons contribute to silver cluster formation. Finally, the effect of microcrystal size on quantum sensitivity is shown in Figure 14 for a fixed trap depth of 0.2 eV. Maximum efficiency is in the 8-10 photons per microcrystal range. These results indicate that the optimum trap density is a function of microcrystal size. This occurs because the significance of each recombination pathway is also a function of microcrystal size. Previous work has shown competition between nucleation and growth to have a negligible effect on the efficiency of formation of three-atom clusters.48 All these simulations assume that electrons and holes are only trapped at the surface. Thus, recombination can only occur at the surface. In larger microcrystals, the hole spends more time wandering through the volume of the microcrystal than it does in small microcrystals. This decreases the free electrodtrapped hole recombination but enhances the free holeltrapped electron recombination. As a result, the trap density required for optimum efficiency shifts toward lower densities as the microcrystal size increases. Although a similar effect also exists for the electron, because of its much higher mobility the effect of microcrystal size is much reduced, and the results in Figure 14 can best be explained by focusing on the effect of microcrystal volume on the partitioning between the trap and free states of the hole. As might be expected, these trends are sensitive to the value of the hole diffusion coefficient.

Discussion and Conclusions The new simulation approach described here is a much more sophisticated simulation than that used previously to study silver cluster formation in silver halide microcrystals. The parameters are those manipulated, or at least potentially manipulated, by those designing new microcrystals and their sensitization for improved photographic properties. The ultimate goal of our

Ag Cluster Formation on AgBr Microcrystals work is a computer-aided design tool. Planned software enhancements and anticipated hardware improvements suggest that this will be possible in the near future. The simulations allow a better understanding of the relationship between the physics of the AgBr microcrystals and their photochemistry. The results show that the formation efficiency of silver clusters can be controlled by optimizing trap density and depth. This is precisely what is done in current practice by optimizing the chemical sensitization of AgBr microcrystals. Development of experimental methods for analysis of the products of chemical sensitization, along with the computer simulation tools described here, will allow the photographic scientist to optimize the microcrystal photochemistry for a given imaging application. There are several improvements needed in the simulation program before it can be applied to microcrystals used in commercial situations. Silver halide microcrystals are known to possess an ionic space ~harge.4~ Due to the ease of formation of interstitial silver ions from surface positive kink sites, there exists an excess negative surface charge balanced by a subsurface layer of interstitial silver ions. The negative surface charge causes an upward bending of the conduction and valence bands. This band bending provides a potential that repels electrons and attracts holes as they enter the subsurface region. Experiments on differently structured microcrystals have suggested that this ionic space-charge potential is not a significant factor in silver cluster formation e f f i ~ i e n c y .Nevertheless, ~~ the effect of the potential needs to be incorporated into the simulation program to understand why it is not important. Silver bromide absorbs light only in the blue region. In order to image light in the green, red, and infrared regions, spectral sensitizing dyes are adsorbed to the microcrystal surface. These dyes provide electrons for silver cluster formation by a photoinduced electron transfer p r o c e ~ s . ~The ' positions of the lowest vacant and highest occupied molecular orbitals of the dye are important features controlling the efficiency of the photoinduced electron transfer. These orbitals also introduce new electron and hole trapping levels at the surface. These effects need to be incorporated into the simulation code so that the process of spectral sensitization may be studied.

Acknowledgment. Some of the software development was carried during my employment at the Eastman Kodak Co. The parallel processor used in our simulations was donated to the Center for Imaging Science by Eastman Kodak Co. I am grateful to Eastman Kodak Co. for both the time and the hardware on which to develop this unique tool. The genesis for the simulation program came from a program developed by Dr. J. Lavine, Eastman Kodak Co., and I am indebted to him for his kind assistance. Finally, J. F. Hamilton laid much of the groundwork in computer simulation of silver cluster formation. His personal mentorship will always be treasured. References and Notes (1) Hamilton, J. F. In The Theory of the Photographic Process; James, T. H., Ed.; Macmillan: New York, 1977; Chapter 4. (2) Hamilton, J. F. Adv. Phys. 1988, 37, 359. (3) (a) Hockney, R. W.; Eastwood, J. W. Computer Simulation Using Particles; McGraw-Hill: New York, 1981; Chapter 10. (b) Snowden, C. M. Semiconductor Device Modelling; Peter Peregrinus: London, 1988. (c) Jacobini, C.; Ligli, P. The Monte Carlo Method for Semiconductor Device Simulation; Springer-Verlag: New York, 1989. (4) Bayer, B. E.; Hamilton, J. F. J. Opt. SOC.Am. 1965, 55, 439. (5) (a) Farnell, G. C.; Chanter, J. B. J. Photogr. Sci. 1961, 9, 73. (b) Hailstone, R. K.; Lieben, N. B.; Levy, M.; McCleary, R. T.; Girolmo, S. R.; Jeanmaire, D. L.; Boda, C. R. J. Imaging Sci. 1988, 32, 113. (6) Seitz, F. Rev. Mod. Phys. 1951, 23, 328.

J. Phys. Chem., Vol. 99, No. 13, 1995 4427 (7) Friauf, R. J. In The Physics of Latent Image Formution in Silver Hailides; Baldereschi, A., Czaja, W., Tosatti, E., Tosi, M., Eds.; World Scientific: Singapore, 1984; pp 79- 116. (8) (a) Hailstone, R. K.; Hamilton, J. F. J . Imaging Sci. 1985, 29, 125. (b) Hailstone, R. K.; Liebert, N. B.; Levy, M.; Hamilton, J. F. J . Imaging Sci. 1987,3I, 185. (c) Hailstone, R. K.; Liebert, N. B.; Levy, M.; Hamilton, J. F. J . Imaging Sci. 1987, 31, 255. (9) Hailstone, R. K.; Hamilton, J. F. J . Imaging Sci. 1987, 31, 229. (10) (a) Hamilton, J. F. In The Physics of Latent Image Formation in Silver Halides: Baldereschi. A,. Czaia. W.. Tosatti. E.. Tosi. M.. Eds.: World Scientific: Singapore, 1984; pp 263-224. (b) Hamilton, J. F. Photogr. Sci. Enn. 1983, 27, 225. (ll)'(a) Hamilton, J. F. Photogr. Sci. Eng. 1974, 18, 371. (b) Harbison, J. M.; Hamilton, J. F. Photogr. Sci. Eng. 1975, 19, 322. (c) Hailstone, R. K.; Liebert, N. B.; Levy, M.; Hamilton, J. F. 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