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THE CONTRIBUTION OF LATTICE MATCHING TO THE INTERFACIAL ENERGY BETWEEN
DISSIMILAR MATERIALS
H. Fecht
To cite this version:
H. Fecht. THE CONTRIBUTION OF LATTICE MATCHING TO THE INTERFACIAL ENERGY
BETWEEN DISSIMILAR MATERIALS. Journal de Physique Colloques, 1988, 49 (C5), pp.C5-171-
C5-176. �10.1051/jphyscol:1988515�. �jpa-00228012�
JOURNAL DE PHYSIQUE
Colloque C5, suppl6ment au nO1O, Tome 49, octobre 1988
THE CONTRIBUTION OF LATTICE MATCHING TO THE INTERFACIAL ENERGY BETWEEN DISSIMILAR MATERIALS
University of Wisconsin, Dept. of Metallurgical and Mineral Engineering, 1509 University Avenue, Madison, W I 53706, U.S.A.
Abstract
The interfacial energy of boundaries between dissimilar materials c a n be described as function of the lattice mismatch, the chemical interaction and the interfacial entropy of the boundaries. Based o n experiments involving a sphere-rotation method and undercooling measurements o r (solid/liquid) phase mixtures i n a droplet dispersion, a n attempt is made to separate t h e influence of the d i f f e r e n t contributions. T h e atomic structure of inter- phase boundaries between noble metals a n d ionic crystals c a n be described by the "lock-in"
model: low energy interphase boundaries were f o u n d if close packed rows of atoms a t the
"surface" of the metal crystal lock into the "valleys" between close packed rows of atoms a t the "surface" of the ionic crystal. At higher temperatures the r e l ~ t i v e stability of d i f f e r e n t interphase boundary structures may change depending on the degree of axial commensuration a n d the related interfacial entropies. Hence, t h e contribution of lattice matching to the interfacial energy can decrease or vanish completely i n some cases, result- ing i n a commensurate/incommensurate phase transition (e.g. f o r Au/A1203). Furthermore, the droplet undercooling experiments demonstrate t h a t good matching between two crystal lattices (substrate/nucleus) c a n f a v o u r formation of metastable phases d u e to the lowering of the activation barrier f o r nucleation during crystallization f r o m a highly undercooled liquid.
Introduction
Considerable attempts have been made i n recent years in developing criteria f o r the correlations between the atomic structure and the energy of internal interfaces. Different geometric models f o r grain boundaries i n metals and oxides have been proposed based on structural unit models i n conjunction with point or planar symmetry 11-31. Due to a limited number of experimental studies of equilibrated boundaries between dissimilar materials this type of interfaces is not well understood. Different experimental approaches including the sphere-rotation method a n d undercooling measurements of liquid droplet samples allow to characterize the properties of interphase boundaries such as the atomic structure, the entropy of t h e interface a n d the chemical (electronic) properties.
Matching a t Metal/Ionic Crvstal Interohase Boundaries
In contrast to grain boundaries i n single component materials, interfaces between metals and ionic crystals a r e formed by a t least three d i f f e r e n t kinds of atoms (or ions).
No computer simulations of the atomic structure of interphase boundaries exist because of the inadequate understanding of specific interatomic potentials f o r these interfaces.
One of the approaches to study the relation between the equilibrium structure a n d the energy of grain or interphase boundaries is offered by t h e sphere-rotation method [4-71. I n this method, about lo7 isolated, single crystal metal spheres (Cu,Ag,Au) of approximately 1 p m diameter a r e sintered i n a n initially metastable arrangement (random
(')present address : Keck Laboratory. California Institute of Technology. Pasadena, CA 91125, U.S.A.
Article published online by EDP Sciences and available at http://dx.doi.org/10.1051/jphyscol:1988515
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PHYSIQUEcrystallographic orientation) onto a f l a t single crystal substrate (LiF, NaCI, KC1, MgO, A1203). By annealing a t elevated temperatures, the spheres minimize their interfacial energy a n d rotate i n t o orientation relationships of low energy. T h e resulting alignment of the crystals can be measured by the conventional methods of X-ray texture analysis. The experimentally observed interphase boundaries of low energy a t intermediate temperatures (550 OC) a r e characterized by parallelism of close packed planes and close packed direc- tions i n both crystals [5]. Moreover, the atomic structure may be described by t h e "lock -in" model. A low energy interphase boundary results if the close packed rows of atoms a t the "surface" of the metal crystal lock into the "valleys" between close packed rows of atoms a t the "surface" of the ionic crystal.
T h i s type of interphase structure has been confirmed by transmission electron microscopy on Au/MgO recently [8] a n d is consistent with results from (relatively simple) computer calculations concerning the adsorption of single metal atoms on the surface of ionic crystals [9-111 using a Van-der-Waals potential. I t was f o u n d t h a t the most stable site f o r a metal atom (Au) is on-top of the cation ( ~ a ' ) of a (100)NaCl or (100)NaF substrate [lo]. Therefore, a high density of "adsorption sites" exists along close packed rows of cations. These directions of maximum adsorption energy a r e parallel to <llO>-direc- tions a t t h e (100)-surface of fcc-crystals a n d have been described a s "valleys" between rows of ions (anions) with weak interaction with the adsorbed atoms [5]. Hence, f o r fcc- metal/fcc-ionic crystal combinations, low energy orientations a r e expected if a maximum number of <IlO>-directions of the metal crystal matches a maximum number of <IlO>-direc- tions a t the "surface" of the ionic crystal resulting i n orientation relationships with
parallel low index planes i n both crystals. T h e required elastic strains to produce lock-in configurations may be reduced by lattice defects 1ike.vacancies a n d dislocations. The interfacial energy (E) f o r crystal combinations with small lattice parameter differences, such as e.g. Au/(OOl)LiF, as function of angle of rotation ((3) around the [IlOI-direction is schematically shown i n Fig. 1. T h e depths of the energy cusps have been taken roughly proportional to t h e number of spheres aligned i n the following low energy orientation relationships:
(1) (OO_l)Au//(OO1)LiF with [ 1 1 0 ] ~ ~ locking i n t o each [ I ~ O ] L ~ F "valley";
(2) (l~l)Au//(OO1)LiF with [ 1 1 0 ] ~ ~ locking into every 7th [11OILiF "valley";
(3) (2z1)Au//(001)LiF with [110lAU locking into every 3d [110ILiF Nvalleyw;
(4) (1 12)Au//(001)LiF with [ I locking into every 5th [ I 1OILiF *valleyu.
Fig. 1: Interfacial energy E of Au/(OOl)LiF interphase boundaries as function of angle of rotation (0) around [llO].
Recently, similar results have been reported [12] by a theoretical treatment of interfacial energies of clusters of u p to 2500 atoms adsorbed a t crystal substrates with cubic structure. Low energy orientation relationships f o r material combinations with varying lattice parameters are obtained if the lattice parameter of both crystals are chosen i n such a way t h a t a high order of axial commensuration is achieved. I n agreement with the "lock-in" model, low energy interphase boundaries a r e obtained if the spacing between close packed rows of the metal atoms (=dM) and the spacinf between the close packed rows of t h e substrate atoms (=dS) retain the relationship ndM = md , where n,m a r e
integers. Consequently, the atomic structure of low energy interphase boundaries between metals a n d ionic crystals c a n be described as a two-dimensional a r r a y of short periodic units of close packed rows of atoms which a r e commensurate i n both crystals a n d f o r m lock-in structures. T h e coincidence site lattice (CSL) model does not appear to apply to interphase boundaries because orientation relationships with low reciprocal density of coincidence sites, C, ( 2 = 5,7,9,11,13 ...) a r e not f o u n d to result i n low energy configura- tions. T h e observed low energy boundaries d o not correspond to high coincidence orienta- tion relationships predicted by the CSL model [5,6].
Metastable Phase Selection due t o Lattice Matchine.
During condensation of metals on semiconductors (e.g. A g / ( l l l ) S i ) a n d metals (e.g.
Fe/(lOO)Cu) structural metastability has been observed frequently [13]. A t low deposition temperatures i t is possible t o grow fully coherent pseudomorphic layers which d u r i n g annealing a t elevated temperatures transform t o their stable form.
Similarly, during crystallization of highly undercooled liquids nucleation of phases with a variety of crystal structures becomes possible. T h e degree of lattice match- ing is expected to influence the selection of the product structures promoting metastable phase formation i n certain cases. T h e most promising approach to study this behavior involves the droplet emulsification technique [14,15]. By dispersing a high purity liquid sample into a large number of small droplets (1-20 diameter) i n a suitable medium, high levels of undercooling (approximately 0.3 t o 0.4 of t h e melting temperature) can be achieved due to a n effective isolation of potent nucleants (Fig. 2).
OIL
MET
Fig. 2: Schematic illustration of nucleant isolation principle i n a droplet dispersion
For some p u r e metals, a variety of crystal phases with d i f f e r e n t lattice struc- tures, has been observed to crystallize during slow cooling of highly undercooled liquids [16,17]. If alloy droplet samples a r e heated i n t o the (solid
+
liquid) two-phase field and subsequently cooled slowly, the metastable liquid in contact with the solid substrate solidifies a t a higher temperature than i n absence of the substrate. Therefore, the influence of the substrate properties on the undercooling level a n d the phase selection during crystallization can be studied systematically. I n Pb-Sn alloys, i t is possible to undercool solid/liquid mixtures (supersaturated a-Pb+
metastable Liquid) u p t o 80 O C below the eutectic temperature 1151. By thermal analysis and "in-situ" X-ray d i f f r a c t i o n methods i t was f o u n d t h a t a metastable crystalline phase with f a c e centered tetragonal structure, PbSn2, nucleates on the&-Pb phase as substrate prior to formation of stable 8-Sn phase.T h e lattice parameters of supersaturated a-Pb, 8-Sn a n d PbSn2 a r e obtained from X-ray d i f f r a c t i o n results. Comparison of the lattice matching of close packed planes of Pb/PbSn2 a n d Pb/@-Sn crystal combinations clearly indicates a better matching of supersaturated a-Pb with metastable PbSn2. Fig. 3a shows the almost epitaxial matching between (111)PbSn2 planes a n d t h e ( 1 I I ) P b substrate, whereas Fig. 3b illustrates poor f i t of 8-(11O)Sn planes a n d (111)Pb. T h e decrease in the activation energy f o r nucleation d u e to the crystallo- graphic compatibility between the nucleus a n d substrate, results in the formation of
C5-174 JOURNAL DE PHYSIQUE
metastable PbSn2 phase rather than the more stable Sn phase.
( 0 ) (b)
Fig. 3: Lattice matching of PbSn2 and &Sn nuclei (N) on a Pb-substrate (S, shaded) favouring metastable phase formation. Fig. 3a: Good matching between metast- able (1 1 1)PbSn2 (dashed) nucleated on (1 1I)Pb. Fig. 3b: Poor matching between
(1 1O)Sn (dot-dashed) a n d (1 11)Pb.
Similarly, i t h a s been f o u n d t h a t i n Sn-alloys, the semiconducting diamond cubic Sn phase (6-Sn) can be synthesized as metastable solidification product f r o m the under- cooled liquid a t temperatures above t h e &-Snlb-Sn transformation temperature if a n appro- priate substrate such as Ge is provided [18]. Again the lattice matching between low index planes is better i n the case of Ge a n d diamond cubic Sn than f o r Ge a n d &(bct)Sn d u e to compatible crystal lattice characteristics, thereby favouring metastable phase formation.
Related effects a n d interfacial e n e r m
I n addition to the lattice matching, other factors such a s temperature (entropy of the boundary), chemical effects a n d the surface topography of a substrate, may contribute to the interfacial energy.
I n most of the purely geometric models f o r the atomic structure of grain or inter- phase boundaries, the influence of temperature on the interfacial energy a n d related interfacial entropy effects are not incorporated. Therefore, measurements of the equili- brium alignment of Au spheres on (100)MgO substrates a t d i f f e r e n t temperatures between 550 and 900 OC were made. T h e results reveal t h a t d i f f e r e n t interphase boundary structures which can be described by the lock-in model [5] have a d i f f e r e n t stability range (see Fig.
4) a s f u n c t i o n of temperature.
T h e crystallographic orientation relationships (I), (3) a n d (4) with high density of locked-in rows of atoms (high axial commensuration) have a high stability over t h e entire temperature range. Whereas the orientation relationship (2) ceases to be a low energy configuration a t temperatures around 900 OC, Therefore, with increasing order of axial commensuration of interfaces, the entropy of the interphase boundary seems to decrease a n d stabilize boundaries with good lattice matching a t high temperatures. Hence, for Au/(lOO)MgO, interphase boundaries with epitaxial type lattice matching (if strain effects can be excluded) a r e expected t o be most stable with respect to temperature.
I . I 500 600 700 800 900
TEMPERATURE ( O C )
Fig. 4: Phase diagram*for Au/(lOO)MgO interfaces indicating the relative number of spheres N / N aligned i n a certain orientation relationship.
In t h e case of Au/A1203 interfaces, it is observed t h a t the ordered structure of interphase boundaries is destroyed a t high temperatures in favour of a more disordered structure. T h e low energy orientation relationships a t 550 OC a r e characterized by parallelism of low index planes ((111)Au//(0001)A120~) a n d low index directions ( [ ~ ~ o ] A u / / [ ~ ~ o o ] A ~ ~ o ~ ) [5] a n d a r e illustrated i n Fig. 5a. Whereas, a t 650 OC the low cnergy orientation is only characterized by parallelism of t h e low index planes irrespec- tive of the relative twist (Fig. 5b).
Fig. 5a: Commensurate Au/A1203 interphase boundary a t 550 OC.
Fig. 5b: Incommensurate Au/A1203 interphase boundary with twist around [ I l l ] above 650 OC.
T h i s transformation f r o m a two dimensional ordered structure (commensurate) to a incommensurate structure with high symmetry has been predicted theoretically 1191 and has been observed experimentally f o r noble gases on graphite [20]. I t appears t h a t the incom- mensurate phase can reduce the strain energy caused by the substrate potential a t higher temperatures by the rotational epitaxy.
In addition, chemical effects a n d the type of bonding between two adjacent crystals c a n influence the interfacial energy. E.g. it is found, t h a t single crystal gold does not grow on vacuum cleaved NaCl [21]. The same e f f e c t of preferred non-epitaxial orientations has been observed f o r Au spheres sintered onto a NaCl substrate. Only a very small number of spheres was f o u n d to be aligned in a n epitaxial orientation [5]. Whereas, f o r Ag on NaCl
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PHYSIQUEboth, vapor deposition methods 1241 and sphere rotation experiments [7] show a strong tendency of a n alignment of the Ag crystals in low interfacial energy orientation relation- ships although the lattice parameters of Ag and Au d i f f e r only about 0.2%. This strong influence of atomic bonding to the interfacial energy might be understood in terms of quantum theoretical approximations, but is difficult to assess a t the moment a n d makes further experiments necessary.
In few cases, i t was observed that during vapor deposition a cube-cube alignment of Sn-or Au-islands [23,24] on amorphous substrates with defined step terraces exist a s low energy orientation. This evidence of a n effect of surface topography on film orientations in a temperature range from 150 to 350 OC can be understood i n terms of preferred nucle- ation sites a t the surface steps and growth along the steps. In the case of the sphere- rotation method, the surface topography can be excluded f r o m the final orientation distri- bution, otherwise the same orientation relationships should be observed independent of the lattice parameter and substrate structure in contrast to the experimental results [5].
In summary, the parameters describing interfacial energies can be separated based on experimental evidence. Important parameters to be considered involve the lattice match- ing between two adjacent phases, the entropy related to the atomic structure of a n inter- phase boundary, chemical effects and, depending on the experimental conditions, the surface topography of a substrate and strains a t the interface. For different crystal combinations the relative contribution of these parameters to the interfacial energy can vary and makes a experimental verification necessary. This will further allow a better understanding of interface related phenomena observed in solid state amorphization, phase transformations a t interfaces a n d the production and stability of metal and semiconductor superlattices.
Acknowledgments
The author likes to thank the Max-Kade Foundation f o r financial support and Prof.
H. Gleiter, Prof. J.H. Perepezko, P. Frankwics and K. Sridharan f o r discussion.
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