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Formation energies of point defects in silver halides : Comparison of atomistically calculated values with values obtained by measurement of surface potentials
F. Granzer, M. Bücher, H.-G. Heuser, P. Petrasch, H. Potstada
To cite this version:
F. Granzer, M. Bücher, H.-G. Heuser, P. Petrasch, H. Potstada. Formation energies of point de- fects in silver halides : Comparison of atomistically calculated values with values obtained by mea- surement of surface potentials. Journal de Physique Colloques, 1980, 41 (C6), pp.C6-101-C6-105.
�10.1051/jphyscol:1980627�. �jpa-00220065�
JOURNAL DE PHYSIQUE Colloque C6, supplkment au no 7 , Tome 41, Juillet 1980, page C6- 101
Formation energies of point defects in silver halides : Comparison of atomistically calculated values with values obtained by measurement of surface potentials
F. Granzer, M. Biicher, H.-G. Heuser, P. Petrasch and H. H. Potstada
I n s u ~ u t lur Angewand~e Physlk der Universitat Frankfurt an Main, Robert-Mayer-Strasse 2-4, F.R.G.
Rksurnk. - Utilisant la correlation forte entre le desordre thermique et les charges d'espace, oppos&es aux charges superficielles, dans les cristaux ioniques, nous avons mesur6 les potentiels superficiels dans AgCl et AgBr en nous servant de la methode de Kelvin-Zisman. Les valeurs obtenues pour les energies deformation des defauts ponctuels (Ag+-interstitiel et Ag+-lacune) ont Cte comparkes avec les valeurs rksultant des calculs atomistiques.
Abstract. - Taking into account the strong interrelation between thermal disorder and surface potentials in ionic crystals, such potentials have been determined for AgCl and AgBr after the Kelvin-Zisman-Vibrating- Capacitor-Method and the results have been-compared with the formation energies of silver ion vacancies and interstitials calculated atomistically.
Introduction. - As first recognized by Frenkel [I]
and later on by Lehovec [2] and Kliewer [3] there exists, in ionic crystals, a strong interconnection between thermal disorder and space- and surface charges or the resulting surface potentials, respective- ly. A systematic study how such surface potentials influence the physical properties of a crystal, being merely of academic interest as far as large alkali halide crystals are concerned, turns out to be of great practical importance if applied to silver halide microcrystals of a photographic emulsion with their high surface-to-volume ratios.
Besides the influence of surface potentials upon the fundamental photographic process in emulsion grains, there is even a further need for reliable values of surface potentials in silver halides : Differing from most of the alkali halides, where formation energies are presently available as a result of atomistic cal- culations [4-51, the corresponding (scarce) data for silver halides, i.e., the formation energies of silver ion vacancies and interstitials, are affected by great uncertainties [5-61. Thus, the direct determination of surface potentials of pure AgCl- and AgBr-crystals can serve as an efficient experimental control for atomistically calculated point defect energies in these crystals.
Following the theory of space charges for ionic crystals, exhibiting thermal disorder of the Frenkel type, the surface potential +s is connected with the free energies of formation of vacancies and intersti- t i a l ~ , F, and Fi respectively, by the equation :
where e is the absolute electronic charge, k the Boltz- mann constant, and T the absolute temperature.
Since the free energy for the formation of a Frenkel defect,
F F = Fi
+
F, , (2) is well known from measurements of ionic conducti- vity it is possible to determine, by experiment, the terms Fi and F , separately when combining (1) and (2) and to compare the corresponding values with those obtained from atomistic calculations. According t6 this procedure, the following (first) part of this report is devoted to the determination of surface potentials of AgX-crystals (I), using the Kelvin- Method, whereas in the second our method of cal- culating point defects in silver halides atomistically will be shortly described and results will be compared with those obtained in part one.1. Determination of surface potentials of AgX- crystals after the Kelvin-Zisman-method. - The only method, allowing the direct determination of sur- face potentials, i.e., the Kelvin-Zisman-Vibrating- Capacitor-Technique, has been used to measure sur- face potentials of thin sheet crystals, bulk Bridgman crystals and evaporated films of AgCl as well as of AgBr.
1 . 1 EXPERIMENTAL. - The electrical operation of the .Kelvin probe circuit, used in our experiments and schematically sketched in figure 1, differs from that described by Danyluk [7]. The guard had been e& = f(Fi - F,) - 4 k T l n 2 , (1) ( I ) The X in AgX means either C1 o r Br.
Article published online by EDP Sciences and available at http://dx.doi.org/10.1051/jphyscol:1980627
C6- 102 F. GRANZER, M. B ~ ~ C H E R , H.-G. HEUSER, P. PETRASCH AND H. H. POTSTADA
The contact potential VAgIAgx between the silver electrode and the AgX-crystal, resulting from the difference of the chemical potentials of a silver ion in silver or in AgX, respectively, is given by the relation :
e. V A ~ I A ~ X = F , A ~ - F A ~ X -
Ampl~fier Band-Amp Amp
- i F , - I + i k T l n 2 + e c p A , , (4)
Schematic Kelv~n Probe Clrcult
where
Ftg
is the free lattice (sublimation) energyFig. 1. - Diagram of the Kelvin-Probe-Circu~l.
separated from the vibrating probe and held fixed so that all undesired signals could be suppressed by means of a differential amplifier.
Figure 2 shows how the different potentials in the Kelvin probe circuit sum up to the measured compensation voltage U n , ,
.
The silver electrode had been chosen as reference potential zero. For more details, concerning the preparation of the AgX- specimens and the arrangement of the vibrating capacitor circuit, see [8].1 . 2 EVALUATION PROCEDURE. - It appears from figure 2, that the AglCu- and Cu!Ag-contact poten- tials cancel each other, so that the measured compen- sation voltage U,,,
,
is given byFtg. 2. - Schematic diagram showing the contact potentials involved in the Kelvin-Method ; I means the Debye-length.
Table I .
per atom and ecp,, the work function, both of metallic silver. I denotes the ionization energy of a silver atom, FF the free energy of formation of a Frenkel pair, already introduced in (2), and F,AgX the free binding energy of a silver ion at a normal lattice site of an AgX-crystal. The latter term, in contrast with all the other terms of (4), which are sufficiently well known from experiments, must be calculated atomistically, as will be shown in part 2.
1 . 3 RESULTS. - Combining (3) and (4) and insert- ing the data collected in table I, where the values for U + in
FtgX
have been anticipated from part 2, one obtains the surface potentials at room temperature, e$,, for AgCl and AgBr, respectively, as listed in the last column. U , is the internal binding energy of a silver atom in Ag, U + the corresponding value for a silver ion in AgX, and UF the formation energy of a Frenkel pair. The appropriate entropies of formation are given by : S + = 6.4 k , So = 0.8 k and SF = 9.4 k or 6.8 k for AgCl or AgBr, respectively. The compen- sation voltage U,,, is the average of measurements performed with a variety of AgX-crystals at room temperature.1 .4 DISCUSSION. - Our surface potentials, e$,, of AgCl and AgBr as delivered by the Kelvin-Zisman- Method are, using our atomistically calculated value U + for the binding energy of a silver ion in AgX, in agreement with most of the results summarized by Tan [9] and they are in excellent agreement with results obtained by Baetzold and Hamilton [lo]
based on surface conductance and diffusion measure- ments. The negative sign of the surface potential for both, AgCl- and AgBr-crystals, indicating the presence of a positive space charge formed of Ag+-interstitials and opposed by a negative surface charge, due to a surplus of negatively charged kink sites, however, is in contradiction to results obtained by Danyluk and Blakely [ l l ] who, likewise using the Kelvin- Method, found a positive surface potential in the case of AgCl.
[eV] F ? g X = U+ - TS+ F? = U o - T S o F F = U F - T S F I t c p ~ , , - 4 s
- - - - - - - -
AgCl 5 . 7 0 = 5 . 8 6 - 0 . 1 6 2.70=2.90-0.20 1 . 2 0 = 1 . 4 4 - 0 . 2 4 7.58 4.73 -0.54 -0.10 AgBr 5 . 6 8 = 5 . 8 4 - 0 . 1 6 2 . 7 0 = 2 . 9 0 - 0 . 2 0 0 . 8 9 = 1 . 0 6 - 0 . 1 7 7.58 4.73 -0.61 -0.30
FORMATION ENERGIES OF POINT DEFECTS I N SILVER HALIDES C6-103
2 . Atomistic calculations of formation energies of point defects in silver halide crystals. - As already mentioned, there still exists a lack of reliable values for the formation energies of silver ion vacancies and interstitials in AgX-crystals in contrast with most of the alkali halides with rock salt 'structure where the formation energies of Schottky defects are now available from atomistic calculations based on the Born-Mayer-Model of ionic crystals. The main reason for this situation, comparing for example NaCl with AgCl, is the electronic configuration, i.e., the 4d-shell of the Ag+-ion. Its deformation, being responsible for the strong violation of the Cauchy- relation, some pecularities in the phonon dispersion curves, and the band structures of AgX-crystals must be taken into account by adding many body terms to the interaction potentials used in atomistic calculations ofpoint defects [12]. In the following, we will separately treat the formation of an Ag+-vacancy and - as a first attempt - the formation of an Ag+-interstitial in both, AgCl and AgBr.
2.1 CALCULATION OF THE FORMATION ENERGY OF A SILVER ION VACANCY. - The energy U for creating an Agf -vacancy is given by :
where Uo is the energy for removing a silver ion from a normal lattice site in an ideal rigid lattice to infinity, - U, the relaxation energy, gained if the lattice is allowed to relax by displacements and electronic polarizations of the ions, and - U, the gain in energy if an Agc-ion is deposited at a kink site. Thus, the binding energy of a silver ion at a normal lattice site, U,, listed in the second column of table I, may be expressed by
Having at hand realistic interaction potentials (see below) there is no problem to calculate Uo and U, because in both cases the crystal is supposed to be ideal. However, the relaxation energy, U,, being a function of the ionic displacements of all ions of the crystal, can only be approximated by a model dividing the crystal in two regions : An inner region, where ionic positions and corresponding energies must be treated atomistically, and an outer region, where the crystal is supposed to behave like a polarizable dielectricurn.
Expressing U, as the difference,
where Uo, and U,, are energies of an AgX-crystal containing an Ag' vacancy before and after relaxation; U,, itself is composed of three parts :
UL is the contribution of ions in the inner region which, in our case, consists of four ionic shells.
U:;" accounts for interactions between the inner and the outer region, both terms, U!, and Uf;': being calculated atomistically. Uz, finally, expresses the response of a dielectric continuum to a point charge as given by the simple formula :
containing the static dielectric constant, E,, and the radius, R, of region I.
The following interaction potentials were used in calculating the terms U& and U,';"of (8) :
Coulomb interaction : U r " ' = qi qj/rij
.
Short range repulsion : U:M = Bij exp(- rij/pij)
.
3-Body (modified Sarkar-Sengupta) potential : Ui3jf = Dijk exp[- (rij
+
rjk)/p3].
Van der Waals-terms : U Z ~ W = - C../y$.
Additional terms, coming from the polarization is the electronic dipole moment induced in the ion i energy per ion i of the inner region, by the monopole and dipole fields EM"" and EDip,
respectively, originating from region I and from the adequate part of region 11. The deformation dipoles pf, supplementing the polarizable point ion model and
-
thus adequately describing the dielectric response,
- - 1 (P; + Pf) dip
,
(I I) are due to the overlapping of electron shells of interact- 2 ing ions and their deformation. All parameters appearing in (10-12) have been adjusted to a variety have been taken into account, where of physical properties like lattice constant and lattice energy, elastic and dielectric constants, and phonon P; = ai(EMo"+
E"P) frequencies in ther
and L points. In order to find theC6-104 F. GRANZER, M. BUCHER, H.-G. HEUSER, P. PETRASCH AND H. H. POTSTADA
Table JI.
l onic displacements
(units of a) U,',,,,, U," U,BM UrvdW U,3B Ur U O Uk U+ = U o - U , U,
- - - - - - - - - - -
= 0.053 AgCl
l 2
= - 0.054t3
= 0.030 2.44 1.35 - 1.85 1.49 0.082 3.51 9.37 4.33 5.86 0.81t 4
= - 0.007t i
= 0.046t2
= - 0.049AgBr
t3
= 0.029 2.19 1.32 - 1.66 1.31 0.073 3.23 9.07 4.88 5.84 0.96r4
= - 0.008final equilibrium configuration around the vacancy, 3. Discussion. - As above mentioned and now the energy of (8) has been minimized with respect drastically evidenced by table 111, the results obtained to the ionic positions in region I. for the formation energies of Ag+-interstitials must be considered to be still preliminary for two reasons : 2 . 2 RESULTS. - The displacements, <,-t,, of
the four inner shells in units of nearest neighbours- distances, are given in the second column of table 11, the negative sign denoting an inward displacement.
The different parts of the interaction energies, contributing to the relaxation energy, U, (column 8), are listed in columns 3-7 and the last column contains the formation energies for Ag+-vacancies.
2.3 CALCULATION OF THE FORMATION ENERGY OF AN INTERSTITIAL S I L V ~ ION. - In principle, the same method has been used here as in the case of an Ag'- vacancy. However, contrary to a vacancy, the inter- stitial neither forms rectangular nor aligned triples with its nearest neighbours. Thus, the 3-body-potential of (10) is no longer applicable and has to be replaced by the Axilrod-Teller potential corresponding to 3-body-Van-der-Waals-interaction,
1
+
3 cos qi cos q j cos q,ui;;
=c.
Z J ~ r; r;, r:, - 2 (12)which is applicable for arbitrary angles p.
Fitting the Axilrod-Teller (AT) parameters to crystal data does not appear as satisfactory as in the case of the Sarkar-Sengupta ,(SS) potentials. Besides this, in computing the interstitial interaction with its neighbouring ions only nearest and next nearest neighbours, for the sake of simplicity, have been considered whereas the fitting procedure even included fourth nearest neighbours.
However, compared with prior results, a trend to realistic values is clearly visible. The improvement has been mainly achieved by replacing the three- body-SS-term of (10) by the AT-potential (12). Thus, if considering, so far, only the atomistically calculated formation energy values of Agf -vacancies, U,, as reliable (last col. of table 11) they may be compared with experimental ones obtained from the surface potentials e+, (last col. of table I) and the formation energies of Frenkel pairs (4 col. of table I) by combin- ing (1) and (2) and dividing the Frenkel pair entropy into equal parts between the vacancy and the inter- stitial :
As appears from table IV, the agreement between 2 . 4 RESULTS. - With the many body potential (12), the theoretical and experimental values for the the energy U y for bringing a silver ion from infinity formation energy of an Ag+-vacancy is excellent to an interstitial position has been calculated (see for AgCl but rather poor in the case of AgBr. The column 2 of table 111). Adding the binding energy, experimental values for the formation energies of U,, of an Ag+-ion with a kink site the formation Ag+-interstitials (last col. table IV) have been obtained energy, U,, of an interstitial is obtained (last col.). using the relation : U,"xP = UF - U y p .
Table 111.
Table IV.
utm
Uk U ,- - - e$, U , U:'P U, U:'p = U F - U
rP
- - - - - -
0.38 AgC1
- 0.10 1.44 0.81 0.81 0.63
0.68 AgBr -0.30 1.06 0.83 0.96 0.23
FORMATION ENERGIES O F POINT DEFECTS I N SILVER HALIDES C6-105
Acknowledgments. - Helpful
,
discussions with The computations were carried out by the help of the Dr. R . Bauer are appreciated. This work has been UNIVAC 1108 of the Hochschulrechenzentrum der supported by the Deutsche Forschungsgemeinschaft. Universitat Frankfurt am Main.Dl SCUSSION
Question. - R. W . WHITWORTH. Comment. - A. B. LIDIARD.
What was the nature of the surface, and does this I think it would be interesting to repeat your mea- affect the experimental or theoretical results ? surements of surface potential at various temperatures and various impurity concentrations, for this could help to verify the theory of defect interactions as described e.g. by Prof. A. R. Allnatt.
Reply. - F . GRANZER. Reply. - F. GRANZER.
Most of the measurements of surface potentials We already studied the influence of temperature have been performed for (100)-surfaces in ultra high and aliovalent cations on surface potentials of silver vacuum. To get reproducible results all crystals were halide crystals. I totally agree with your comment that heated up to temperatures near the melting point in from a comparison of surface potentials of pure and order to get clean surfaces. doped AgCl- and AgBr-crystals one gets information
on the interaction of the various defects.
References
[I] FRENKEL, J., Kinetische Theorie der Flussigkeiten (VEB, Deutscher Verl. d. Wiss., Berlin) 1957.
[2] LEHOVEC, K., J. Chem. Phys. 21 (1953) 1123.
[3] KLIEWER, K. L., J. Phys. Chem. Solids 27 (1966) 705.
141 CATLOW, C. R . A., CORISK, J., DILLER, K. M., JACOBS, P. W. M.
and NORGETT, M. J , Harwell Report TP 713 (1977).
[5] LEUTZ, K., Thesis, Stuttgart (1977).
[6] GRANZER, F., BELZNER, V., BUCHER, M., PETRASCH, P. and
POTSTADA, H., Int. Conj. on Defects in Insul. Cryst., Gatlinburg (1977).
[7] DANYLUK, S., J. Phys. E (Sci. Instr.) 5 (1972) 478.
[8] HEUSER, G., Thesis, Frankfurt am Main (1979).
[9] TAN, YEN, T., Prog. Solid State Chem. 10 (1976) 193.
[lo] BAETZOLD, R. C. and HAMILTON, J. F., SurJ Sci. 33 (1972) 461.
[ l l ] DANYLUK, S. and BLAKELY, J. M., SurJ Sci. 41 (1974) 359.
[12] BUCHER, M., Thesis, Frankfurt am Main (1978).
