Thermodynamics of reactions and phase transformations at ...

L. P. H. Jeurgens et al.: Thermodynamics of reactions and phase transformations at interfaces and surfaces

Lars P. H. Jeurgens,a Zumin Wang,a Eric J. Mittemeijera,b a b

Max Planck Institute for Metals Research, Stuttgart, Germany Institute for Materials Science, University of Stuttgart, Germany

Thermodynamics of reactions and phase transformations at interfaces and surfaces Recent advances in the thermodynamic description of reactions and phase transformations at interfaces between metals, semiconductors, oxides and the ambient have been reviewed. Unanticipated nanostructures, characterized by the presence of phases at interfaces and surfaces which are unstable as bulk phases, can be thermodynamically stabilized due to the dominance of energy contributions of interfaces and surfaces in the total Gibbs energy of the system. The basic principles and practical guidelines to construct realistic, practically and generally applicable thermodynamic model descriptions of microstructural evolutions at interfaces and surfaces have been outlined. To this end, expressions for the estimation of the involved interface and surface energies have been dealt with extensively as a function of, e. g., the film composition and the growth temperature. Model predictions on transformations at interfaces (surfaces) in nanosized systems have been compared with corresponding experimental observations for, in particular, ultrathin (< 5 nm) oxide overgrowths on metal surfaces, as well as the metal-induced crystallization of semi-conductors in contact with various metals. Keywords: Interface energy; Surface energy; Thermodynamics; Transformations; Nanomaterials; Thin films; Amorphous solids.

1. Introduction The thermodynamics of reactions and phase transformations in nanomaterials, with their characteristically high interface Int. J. Mat. Res. (formerly Z. Metallkd.) 100 (2009) 10

density, can deviate significantly from “expected” behaviours for “bulk” materials, e. g. as derived from “bulk” phase diagrams [1 – 4]. Thin film systems (with (sub)layer thicknesses in the nanometer range) provide classical examples of such nanomaterials. Obviously, the relatively large volume fractions of atoms associated with interfaces and surfaces in such low-dimensional systems (i. e. with one dimension, the layer thickness, expressed in the size scale of atoms) can bring about energy contributions, which activate mechanisms for microstructural changes, which are insignificant in corresponding “bulk” systems. It should be recognized that the relatively high volume fraction of material at interfaces (and surfaces) in a nanosized system not only has pronounced consequences for the energetics of the system: dimensional and microstructural constraints occur by disturbing the lattice periodicity, thereby confining the mobilities of e. g. photons, phonons, plasmons and/or dislocations [5 – 7]. In general, low-dimensional systems, such as thin films and sheet assemblies (2-dimensional systems); wires, tubes, chains and rods (1-dimensional systems); nano-particles and quantum dots (0-dimensional systems), as well as nano-grained polycrystalline materials, exhibit properties that differ significantly from their corresponding bulk materials: e. g. a much higher yield strength, a strikingly lower or higher melting point (i. e. premelting or superheating behavior, respectively) and/or specific electrical, magnetic and optical properties [4 – 10]. In this review the focus is on the thermodynamic properties of interfaces and surfaces and their consequences for nanomaterials, such as thin film systems. Typical thermodynamic driving forces for microstructural transitions in thin film systems in contact with the am1281

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Feature L. P. H. Jeurgens et al.: Thermodynamics of reactions and phase transformations at interfaces and surfaces

bient (i. e. vacuum, a gas atmosphere or an adsorbed layer) are the lowering of surface, interface, and/or grain-boundary energies by (impurity) segregation effects [11, 12], adsorbate-induced surface reconstructions [13 – 15], wetting [4, 16 – 18] and/or interfacial mixing or compound formation [1, 2, 19] (often resulting in metastable crystalline or amorphous interfacial or surficial layers). Furthermore, the relatively short diffusion distances (normal to the film surface) in combination with the large volume fraction of material associated with surfaces and interfaces, which both can act as fast diffusion paths, enable much faster kinetics for thermodynamic equilibration, which makes artificially, man-made thin film systems prone to degradation. Controlling the thermodynamic stability of solid–solid interfaces between metals, alloys, semiconductors, oxides, biomaterials, and the ambient is therefore of cardinal importance in numerous state-of-the-art nano-technologies, such as those to produce novel structural materials based on metal/ceramic composites [20 – 22], metal/oxide seals in device and medical implant construction [21, 22], metal/oxide contacts in microelectronics and photovoltaic devices [23 – 25], coatings for corrosion resistance [22, 26, 27], gas-sensors [28, 29] and oxide-supported transition metal catalysts [30, 31]. In recent years, important achievements have been made in the theoretical description of microstructural evolutions at contacting interfaces and at surfaces, on the basis of interface thermodynamics, thereby invalidating the frequently applied, evasive invocation of a “kinetic” constraint to “understand” the experimental observation of unanticipated (nano)structures, which differ from those known and predicted by bulk thermodynamics. For example, experimental observations of the formation of amorphous alloy phases by interdiffusion at interfaces and grain boundaries of crystalline multilayers in e. g. the Ni–Ti, Cu–Ta, Al–Pt, and Mg–Ni system (a process commonly referred to as solid-state amorphisation; SSA) have previously been rationalized on the basis of criteria which involve kinetic hindrance of the formation of a corresponding crystalline intermetallic compound; such criteria focused on the large atomic size mismatch of the constituents and/or the “anomalously” fast diffusion of one of the constituents (see the references listed in Refs. [1, 32, 33]). However, such thinking is erroneous: recent thermodynamic model predictions [1, 2] demonstrate that the energy of the interface between an amorphous phase and a crystalline phase is in many cases lower than that of the corresponding crystalline–crystalline interface. Consequently, thin amorphous films developing at the interface (surface) and/or grain boundaries can in principle be thermodynamically stable up to a certain critical thickness, as long as the higher bulk energy of the amorphous phase (as compared to the competing bulk crystalline phase of the same composition) is overcompensated by its lower sum of the crystalline–amorphous interface (surface) energies [1, 2]. By now, many experimental observations of stable intergranular and/or surficial amorphous films at ceramic– ceramic and metal–metal grain boundaries, ceramic–ceramic heterointerfaces and metal–oxide interfaces (with typical equilibrium thicknesses in the range of 1 to 2 nm) have been successfully explained on such a thermodynamic (rather than a kinetic) basis: see Refs. [1 – 4, 34 – 39] and references therein. 1282

To satisfy the technological demand for control of the thermodynamic stability and related properties of low-dimensional functional systems under operating conditions, versatilely applicable and accurate thermodynamic model descriptions are needed for the energetics of the contacting interfaces between (and surfaces of) the various system components. Driving forces for reactions and phase transformations at interfaces and surfaces should be modeled as function of the material and operating conditions, such as the film thickness, the chemical composition and constitution, the temperature and operation time, as well as the ambient conditions. Up to date, such generally applicable descriptions of, in particular, solid–solid interfacial energies can only be assessed (readily and) successfully for practical application by semi-empirical expressions as derived on the basis of the macroscopic atom approach [1 – 3, 40 – 43], originally proposed and developed by Miedema and co-workers [44 – 46] (see Sections 2 and 4 for details). The present paper provides a detailed overview of recent accomplishments in the aforementioned modeling of reactions and phase transformations in nanomaterials as thin film systems on a thermodynamic basis by accounting for the crucial role of interface and surface energies. The design of a thermodynamic model, specifying the essential energy contributions, is discussed and illustrated in Section 2. The needed expressions for the assessment of solid surface Gibbs energies of amorphous and crystalline metals, semiconductors, and oxides as a function of the temperature are provided in Section 3. Expressions for the estimation of the Gibbs energies of heterointerfaces between crystalline and amorphous metals, semiconductors and oxides, as a function of the temperature and the size of the system (as given by the film thickness) are presented in Section 4. Finally, comparison of thermodynamic predictions, on the above basis, with experimental observations is presented in Sections 5 and 6 regarding: (i) the relative stabilities of ultrathin (< 5 nm) amorphous and competing crystalline oxide overgrowths on bare metal substrates, and (ii) the crystallization of amorphous semiconductors at the interfaces with various adjoining metals at temperatures well below their bulk crystallization temperature (a process commonly referred to as metal-induced crystallization).

2. Basis of thermodynamic analysis; identification of energy contributions Thermodynamic analysis of a phase transformation begins with the identification and evaluation of the involved driving force(s). The total Gibbs energy change of a system accompanying a phase transformation from state A to state B tot ¼ GBtot GAtot (Fig. 1). The thermodyis given by: DGA!B namic driving force is defined as the negative of the total tot , i. e. a positive driving Gibbs energy change: DGA!B force exists if the phase transformation is associated with a lowering of the system’s total Gibbs energy (i. e. tot ¼ GBtot GAtot < 0). A thermodynamically desired DGA!B tot ¼ GBtot GAtot < 0) can be phase transformation (DGA!B hindered by kinetic barriers expressed by a single rate-limiting activation energy, as QA!B in Fig. 1, or by several (a series of) rate-determining [47, 48]) activation energies, which may be overcome by thermal activation. For examInt. J. Mat. Res. (formerly Z. Metallkd.) 100 (2009) 10

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Fig. 1. Energy change of a system accompanying a transformation from state A to state B. The driving force is expressed by the negative of the decrease of the system’s Gibbs energy accompanying the transtot formation: i. e. DGB!A ¼ GBtot GAtot . The transformation can be kinetically hindered at low temperatures by the associated activation energy barrier, QA!B .

ple, in the case of an amorphous-to-crystalline solid-state phase transformation [3, 18], the thermal energy of the involved atoms can be too low to move (i. e. diffuse) and/or to rearrange themselves to form a particle of critical size (nucleus) of the thermodynamically preferred bulk crystalline phase. Or, analogously, the solid-state wetting of grain boundaries (as driven by a lowering of the total grainboundary energy) can be thermally activated according to the energy barrier for grain-boundary diffusion [16, 17]. The driving forces for phase transformations in materials of high interface density as low-dimensional systems (i. e. with interface distances of the order of nm, i. e. expressed by a distance scale of the size of atoms) are no longer governed by the associated changes in bulk energy of the solid, but can instead be predominated by the accompanying changes in the surface and interface energies [1 – 4, 17, 41, 43]. For example, the solid-state transformation from an amorphous to the corresponding crystalline state is always preferred by bulk thermodynamics, but is often counteracted in thin film systems by energy penalties for the creation of crystalline surfaces and crystalline/crystalline interfaces from the original amorphous surfaces and crystalline/amorphous interfaces, respectively (see Section 4.2. and e. g. Refs. [1 – 4, 41]).

The thermodynamic basis needed to arrive at realistic model descriptions of solid-state phase transformations (or reactions) at interfaces, as in thin film systems, involves specification of all energy contributions to a phase transformation. This non-trivial set-up of an appropriate thermodynamic model is illustrated in the following by two examples: (i) the formation of a binary solid solution at the interface between two crystalline metals and (ii) the development of an overgrowth of an ultrathin (< 5 nm) oxide film on a bare (i. e. without a native oxide) metal substrate by thermal oxidation. 2.1. Solid-solution formation at interfaces Consider the formation of a binary AB solid solution by interdiffusion at the interfaces of a binary A-B multilayer upon annealing. The bulk-thermodynamic driving force for the formation of a crystalline hABi solid solution at the hAijhBi interfaces is provided by a negative Gibbs energy of mixing of the two crystalline components hAi and hBi.1 The angle brackets, h i, are used to denote a crystalline phase. Adopting the treatment in Ref. [1], the thermodynamic analysis of such a phase transformation will be described for a unit cell of volume ½ hhAi þ hhBi per unit interface area, as defined in Fig. 2a, where hhAi and hhBi are the initial layer thicknesses of hAi and hBi, respectively. The total Gibbs energy of the defined unit cell for the initial state of the A-B multilayer (i. e. before reaction) is given by cell ¼ hhAi GhAijhBi

GhAi GhBi þ hhBi þ 2 chAijhBi VhAi VhBi

ð1Þ

where GhAi and GhBi are the Gibbs energies (per mole) and VhAi and VhBi the molar volumes of hAi and hBi, respectively, and chAijhBi denotes the interface energy of the hAijhBi interface (per unit area). After formation of thin pro1 The formation of a crystalline intermetallic compound will not be considered in the present treatment.

Fig. 2. Schematic drawing of a binary A–B multilayer before and after formation of hABi solid-solution product layers at the original interfaces by interdiffusion. The thermodynamics of formation of crystalline and amorphous solid-state product layers at the original interfaces are calculated for the indicated unit cell (as defined per unit interface area) with a height equal to the sum of the thicknesses hhAi and hhBi of the initial crystalline hAi and hBi metal layers, respectively. See Section 2.1. for details.

Int. J. Mat. Res. (formerly Z. Metallkd.) 100 (2009) 10

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duct layers of the crystalline hABi solid solution at the original hAijhBi interfaces (with uniform thicknesses, hhABi ), the total Gibbs energy of the defined unit cell is given by (Fig. 2b) GhAi cell GhAijhABijhBi ¼ ðhhAi hhABi!hAi Þ þ ðhhBi hhABi!hBi Þ VhAi

GhBi GhABi þ ðhhABi!hAi þ hhABi!hBi Þ þ2 VhBi VhABi

ðchAijhABi þ chBijhABi Þ

ð2aÞ

where 12 hhABi!hAi and 12 hhABi!hBi are the thicknesses of the hABi layers grown in layer hAi and hBi, respectively (with hhABi ¼ 12 hhABi!hAi þ 12 hhABi!hBi ; see Fig. 2b; GhABi and VhABi represent the Gibbs energy and molar volume of the hABi solid solution; and chAijhABi and chBijhABi denote the interface energies per unit area of the hAijhABi and hBijhABi interfaces, respectively. The Gibbs energy of formation of 1 mole of hABi out of its elements in their bulk stable configuration is given by: f DGhABi GhABi xA GhAi ð1 xA Þ GhBi

ð2bÞ

where xA denotes the mole fraction of A in hABi. Provided that the molar volumes of the components hAi and hBi do not significantly change upon alloying (i. e. VhABi ¼ xA VhAi þ ð1 xA Þ VhBi ), Eq. (2a) can be rewritten as f DGhABi GhAi GhBi cell GhAijhABijhBi ¼ hhAi þ hhBi þ 2 hhABi VhAi VhBi VhABi þ 2 ðchAijhABi þ chBijhABi Þ

ð2cÞ

Hence the Gibbs energy change, DG CSS , upon formation of crystalline solid solution (CSS) product layers at the hAijhBi interfaces of the original A-B bilayer is given by cell cell DG CSS ¼ GhAijhABijhBi GhAijhBi ¼ 2 hhABi

f DGhABi

VhABi

þ 2 ðchAijhABi þ chBijhABi chAijhBi Þ

ð3aÞ

If an amorphous {AB} (instead of a crystalline hABi; the braces, fg, are used to denote an amorphous phase) solid solution product layer is formed at the original hAijhBi interfaces (a process commonly referred to as solid-state amorphisation; SSA) a similar expression for the associated Gibbs energy change, DGSSA , results: f DGfABg cell cell GhAijhBi ¼ 2 hfABg DGSSA ¼ GhAijfABgjhBi VfABg þ 2 ðchAijfABg þ chBijfABg chAijhBi Þ

ð3bÞ

f where hfABg , DGfABg , and VfABg denote the total thickness, the Gibbs energy of formation and the molar volume of the amorphous {AB} product layer, respectively; chAijfABg and chBijfABg are the energies of the interfaces per unit area between the amorphous {AB} phase and the crystalline hAi and hBi components, respectively. In the derivation of Eq. (3b), the molar volume of the amorphous {AB} solid solution is taken the same as that of the corresponding crystalline hABi solid solution of identical composition (i. e. VfABg ffi VhABi ).

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The driving forces for the formation of crystalline (i. e. DG CSS according to Eq. (3a)) and amorphous (DG SSA according to Eq. (3b)) solid solution product layers at the original hAijhBi interfaces can now be evaluated as a function of the annealing temperature (T) and the composition of the A-B product phase (as expressed by the molar fraction, xA ), provided that corresponding expressions for the bulk Gibbs f f and DGfABg ) and the enerenergies of formation (i. e. DGhABi gies of the interfaces between the various crystalline and amorphous phases (chAijhBi , chAijhABi , chBijhABi , chAijfABg , and chBijfABg ) are assessable as a function of T and xA . Experimenf tal values and procedures for the assessment of DGhABi and f DGfABg as a function of T and xA are provided in, e. g., Refs. [1, 43, 44, 46, 49 – 51]. Generally applicable formulations for the calculation of the crystalline–amorphous and crystalline–crystalline interface energies as a function of T and xA are presented in this paper (Sections 4.2.1. and 4.3.1., respectively). An example of such thermodynamic model predictions for the occurrences of crystalline and amorphous solid solution product layers at the interfaces of a Ni–Ti multilayer by interdiffusion at 525 K (as calculated for a unit cell of lateral area of 10 · 10 nm2 and with individual layer thicknesses of 10 nm; see Fig. 2) is provided by Fig. 3. The calculated interface energies, chNiijhNiTii , chTiijhNiTii , chNiijfNiTig , and chTiijfNiTig (per unit interface area (symbol c), as well as per volume of the defined unit cell (symbol C); see Sections 4.2.1. and 4.3.1.) have been plotted as a function of xNi in Fig. 3a. The bulk Gibbs energies of formation, f f DGhNiTii and DGfNiTig are shown (also as a function of xNi ) in Fig. 3b. It follows that the crystalline–amorphous interface energies (i. e. chNiijfNiTig and chTiijfNiTig ) are always lower than the corresponding crystalline–crystalline energies (chNiijhNiTii and chTiijhNiTii ) (Fig. 3a). Consequently, the amorphous {NiTi} product layer can be thermodynamically preferred with respect to the corresponding crystalline hNiTii product layer, as long as the energy penalty due to the higher bulk energy of the amorphous solid solution (i. e. f f > DGhNiTii ; see Fig. 3b) is overcomless negative: DGfNiTig pensated by its relatively lower sum of crystalline–amorphous interface energies (i. e. ½chNiijfNiTig þ chTiijfNiTig < ½chNiijhNiTii þ chTiijhNiTii ). Thus the model predicts a distinct positive driving force for interface amorphisation in Ni-Ti multilayers (see Fig. 3c), in accordance with experimental observations [32]. crit A theoretical value for the critical thickness, hfNiTig , up to which the amorphous fNiTig product layer is thermodynamically, rather than kinetically (cf. Section 1), preferred (as compared to the competing crystalline hNiTii product layer of identical composition) is obtained by solving CSS SSA DGhNiTii ðhhNiTii ; xNi ; TÞ ¼ DGfNiTig ðhfNiTig ; xNi ; TÞ for hfNiTig for a given mole fraction, xNi , of Ni in NiTi (employing Eqs. (3a) and (3b) with VfNiTig ffi VhNiTii ; see above): crit hfNiTig ðxNi ; TÞ

¼

ðchNiijhNiTii þ chTiijhNiTii Þ ðchNiijfNiTig þ chTiijfNiTig Þ f f DGhNiTii Þ=VfNiTig ðDGfNiTig

ð4Þ

crit For hfNiTig > hfNiTig , bulk energy contributions become dominant and the product layer will strive for crystallization as a crystalline solid solution (or as a crystalline intermetallic compound). The calculated critical thickness,


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crit hfNiTig , has been plotted as a function of xNi in Fig. 4. It folcrit has a maximum value of about 23 nm lows that hfNiTig for xNi = 0.56 (at 525 K). The theoretical value of crit *10 nm for xNi = 0.50 agrees very well with the exhfNiTig perimentally determined thickness of 9.0 ± 0.2 nm of the amorphous Ni50Ti50 product layer formed after annealing of an as-prepared 10 nm-Ni/16 nm-Ti multilayer for 720 min at 523 K (as observed by HRTEM) [52].

(a)

2.2. Oxide formation at metal surfaces

(b)

(c)

Fig. 3. Thermodynamic predictions for the formation of amorphous and crystalline solid solution product layers at the original hNiijhTii interfaces of a Ni–Ti multilayer by interdiffusion at 525 K, as calculated for individual layer thicknesses of hhNii = hhTii = 10 nm and a unit cell of lateral area 10 · 10 nm2 and (see Fig. 2). (a) The calculated crystalline–crystalline and crystalline–amorphous interface energies as function of the Ni fraction, xNi, in the Ni–Ti product layer (see Sections 4.3. and 4.2.1., respectively). (b) Bulk Gibbs energies of formation of the amorphous fNiTig and crystalline hNiTii product phases as a function of xNi. (c) The calculated free energy change for solid-state amorphization at the interfaces of the Ni–Ti multilayer as a function xNi (occuring as depicted in Fig. 2; see Section 2.1.). The ordinates at the left- and right-hand sides of pannels (a) and (b) give the corresponding energies per unit-cell volume and per unit interface area, respectively [1].

Consider the formation of either an amorphous or a crystalline oxide overgrowth on a bare (i. e. without a native oxide) single-crystalline metal substrate, hMi. The energetics of an amorphous oxide overgrowth, {MxOy}, with uniform thickness, hfMx Oy g , can be compared with those of the competing crystalline oxide overgrowth, hMx Oy i, of equivalent uniform thickness, hhMx Oy i [2]. Again the braces, { }, and angle brackets, h i, refer to the amorphous state and the crystalline state, respectively. The competing fMx Oy g and hMx Oy i oxide films are grown from the same molar quantity of oxygen reactant on identical metal substrates at the same growth temperature, T. The total Gibbs energies of the concerned hMijfMx Oy g and hMijhMx Oy i configurations will be compared for unit 2 2 cells of volumes hfMx Oy g lfM and hhMx Oy i lhM , rex Oy g x Oy i spectively, such that the defined unit cells contain the same molar quantity of oxygen reactant (and thus the same molar quantity of oxide phase, provided that the compositions of the competing oxide overgrowths are identical); see schematic drawings in Fig. 5. The accumulation of elastic growth strain in the amorphous fMx Oy g overgrowth (and the metal substrate) can be neglected due to the relative large free volume and moderate bond flexibility of the amorphous structure (see Refs. [2, 53, 54] and references therein; see also Section 4.2.2.). For the (semi-)coherent crystalline hMx Oy i overgrowth, on the other hand, the initial lattice mismatch between hMx Oy i and hMi, which is governed by the crystallographic orientation relationship (OR) of the (semi-)coherent hMx Oy i overgrowth with the parent metal substrate (see Section 4.3.), generally leads to the build up of a planar state of (tensile or compressive) residual growth strain [2, 41]. Since the unstrained fMx Oy g and strained hMx Oy i unit cells contain the same molar quantity of oxygen reactant, it holds that 2 lfM hfMx Oy g x Oy g

VfMx Oy g

crit Fig. 4. Calculated critical thickness, hfNiTig , up to which an amorphous fNiTig solid solution product layer is thermodynamically preferred with respect to the corresponding crystalline hNiTii solid-solution product layer as a function of the molar fraction of Ni for interface amorphisation in Ni–Ti multilayers at 525 K [1].


¼

2 lhM hhMx Oy i x Oy i

XhMx Oy i VhMx Oy i

ð5aÞ

where VfMx Oy g and VhMx Oy i are the molar volumes of strainfree fMx Oy g and strain-free hMx Oy i, respectively; lfMx Oy g and lhMx Oy i correspond to the widths and lengths in perpendicular directions along the interface plane of the unstrained fMx Oy g unit cell and the strained hMx Oy i unit cell, respectively (see Fig. 5). The fraction XhMx Oy i on the right-hand side str , as occupied by one of Eq. (5) relates the volume, VhM x Oy i mole Mx Oy in the strained hMx Oy i overgrowth, to the molar str volume of strain-free hMx Oy i: i. e. VhM ¼ XhMx Oy i x Oy i VhMx Oy i . Furthermore, the corresponding ratios of the heights (i. e. thicknesses) and surface areas of the unstrained fMx Oy g cell and strained hMx Oy i cell are given 1285

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by [2, 41] hhMx Oy i n¼ ¼ ð1 þ e33 Þ hfMx Oy g

VhMx Oy i VfMx Oy g

!1 3

ð5bÞ

and v¼

lhMx Oy i lfMx Oy g

!2

VhMx Oy i ¼ ð1 þ e11 Þ ð1 þ e22 Þ VfMx Oy g

!2 3

ð5cÞ

respectively, where eij represents the residual, homogeneous strain tensor of the hMx Oy i overgrowth with the corresponding perpendicular directions 1 and 2 parallel to the hMijhMx Oy i interface plane and the direction 3 perpendicular to the interface plane (for details, see Section 4.3.2. and Ref. [41]). The total Gibbs energies of the defined unit cells of the amorphous and crystalline oxide overgrowths are given by (Fig. 5a) cell 2 ¼ lfM GfM x Oy g x Oy g

hfMx Oy g

GfMx Oy g S þ cfM x Oy g VfMx Oy g !

þ chMijfMx Oy g

ð6aÞ

and (Fig. 5b) cell 2 ¼ lhM GhM x Oy i x Oy i

hhMx Oy i

GhMx Oy i S þ chM x Oy i XhMx Oy i VhMx Oy i !

þ chMijhMx Oy i

ð6bÞ

where GfMx Oy g and GhMx Oy i are the bulk Gibbs energies per S S mole fMx Oy g and hMx Oy i; cfM and chM represent x Oy g x Oy i the surface energies (per unit area) of the fMx Oy g and hMx Oy i overgrowths in contact with the ambient (e. g. vacuum, a gas atmosphere or an adsorbed layer); and chMijfMx Oy g and chMijhMx Oy i are the energies (per unit area) of the interfaces between the metal substrate and the fMx Oy g and hMx Oy i oxide overgrowth, respectively. The Gibbs energy of formation, DGMf x Oy , of one mole Mx Oy oxide phase out of its elements in their stable configuration, at a given temperature and pressure, is defined as y f GfMx Oy g x GhMi GO2 ðgÞ DGfM x Oy g 2

ð7Þ

By employing Eqs. (5a), (5c), (6a), (6b), and (7), it then follows that the thermodynamic stability of the amorphous oxide overgrowth with respect to that of the competing crystalline oxide overgrowth, as expressed by the difference in total Gibbs energy of the corresponding fMx Oy g and hMx Oy i unit cells (Fig. 5), is given by ! f f DGfM DGhM x Oy g x Oy i cell cell cell DG ¼ GfMx Oy g GhMx Oy i ¼ hfMx Oy g VfMx Oy g S S þ cfM þ chMijfMx Oy g v ðchM þ chMijhMx Oy i Þ ð8Þ x Oy g x Oy i

Thus if DG cell ðhfMx Oy g ; TÞ < 0 the amorphous oxide over1286

(a)

(b) Fig. 5. Schematic drawing of competing amorphous and crystalline oxide overgrowths of uniform thicknesses (< 5 nm) on top of their bare, single-crystalline metal substrates, hMi, in contact with the ambient (e. g., vacuum, a gas atmosphere or an adsorbed layer). (a) the homogeneous amorphous oxide overgrowth, fMx Oy g, of uniform thickness, hfMx Oy g , on the metal substrate. (b) the competing crystalline oxide overgrowth, hMx Oy i, of uniform thickness, hh Mx Oy i , on the same metal substrate. The competing amorphous and crystalline oxide phases have the same composition and were formed from the same molar quantity of oxygen reactant on identical single-crystalline metal substrates. 2 and Furthermore, the defined unit cells of volume hfMx Oy g lfM x Oy g 2 hhMx Oy i lhMx Oy i , as indicated in (a) and (b) contain the same molar quantity of fMx Oy g and hMx Oy i, respectively.

growth is more stable, whereas for DG cell ðhfMx Oy g ; TÞ > 0 the (strained) crystalline oxide cell is more stable. Evidently, for thick oxide overgrowths, the bulk energetic contributions will stabilize the crystalline oxide overgrowth (since f of a crystalline oxide phase will always be lower DGhM x Oy i f of the corresponding amorphous oxide phase than DGfM x Oy g [55]). For very thin oxide overgrowths, the higher bulk Gibbs energy of the amorphous oxide phase can in principle be overcompensated by its lower sum of surface and interface energies (as compared to the corresponding crystalline oxide configuration). Hence, the amorphous oxide overgrowth can crit be stable up to a certain critical thickness, hfM [2, 3, 41, 56]. x Oy g A theoretical prediction of this critical thickness, up to which the amorphous oxide overgrowth on the bare metal is thermodynamically, rather than kinetically (cf. Section 1), preferred (as compared to the competing crystalline overgrowth on the same metal), is obtained by solving hfMx Oy g in Eq. (8) for DG cell ðhfMx Oy g ; TÞ = 0 [2, 3, 41]. To this end, versatilely applicable formulations for the bulk, surface, and interface energy terms of the fMx Oy g and hMx Oy i overgrowths, as a function of the oxide-film thickness, growth temperature, metal substrate orientation, and its OR with the (semi-)coherent hMx Oy i overgrowth are required. These are presented in Sections 3.2., 4.2.2., and 4.3.2. If the resulting value of critical is negative, it is implied that the oxide overgrowth hfM x Oy g Int. J. Mat. Res. (formerly Z. Metallkd.) 100 (2009) 10

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on the bare metal substrate is thermodynamically predicted to proceed by the direct formation and growth of a (semi-)coherent crystalline oxide phase. The calculated bulk, interface, and surface energy differences, as well as the corresponding total Gibbs energy differcell cell GhM , (see corresponding terms ence, DG cell ¼ GfM x Oy g x Oy i in Eq. (8)) for competing amorphous and crystalline MgO, TiO2 and SiO2 overgrowths on their bare hMg{0001}i, hTi{0001}i and hSi{111}i substrates at a growth temperature of T = 298 K, are plotted in Fig. 6 as function of oxide-film thickness in the range of 0 £ hfMx Oy g £ 3 nm. The corresponding critical thicknesses up to which the amorphous {MgO} and {TiO} overgrowths are thermodynamically preferred on their most densely packed metal substrates follow from the intercepts of the calculated DG cell -curves with the abscis-

crit sa in Fig. 6a and b, respectively; i. e. hfMgOg % 0.2 nm and crit hfTiO2 g * 0.8 nm. Amorphous {SiO2} is the preferred initial oxide overgrowth far beyond the largest film thickness concrit > 40 nm [3]), sidered in the present calculations (i. e. hfSiO 2g in accordance with experiment [57]. For oxide overgrowths on hMg{0001}i, the calculated critical oxide-film thickness is below 1 oxide monolayer (ML*0.22 nm) [3], which indicates that the development of a, thermodynamically stable, amorphous oxide film on bare Mg{0001} metal surfaces is unlikely. The results of these critical thickness calculations are compared in further detail with experimental data in Section 5.1.

3. Assessment of solid surface energies The Gibbs energy of surface atoms is increased with respect to that of the corresponding bulk atoms due to the deficient state of chemical bonding at a liquid or solid surface. The surface energy, cAS , (per unit area) of a homogeneous solid phase A at temperature T can then be defined as the excess Gibbs energy of the constituent surface atoms or molecules of A (relative to bulk atoms or molecules of A) per unit surface area, i. e. cAS ðTÞ ¼

(a)

HAS ðTÞ T SAS ðTÞ OA ðTÞ

ð9aÞ

where HAS and SAS are the excess enthalpy and the excess entropy of the defined system due to the presence of the surface OA in the defined system.2 The temperature dependence of the surface energy, qcAS =qT (e. g. in J m – 2 K – 1), is mainly governed by the surface entropy and thermal expansion (or shrink) [44, 58, 59]. Consequently, neglecting the individual temperature dependencies of HAS , SAS and OA , the resulting temperature dependence of the surface energy, qcAS =qT, is given by qcAS =qT ffi SAS =OA

(b)

(c)

Fig. 6. Calculated bulk, interfacial and surface energy differences, as well as the corresponding total Gibbs energy difference cell cell (DG cell ¼ GfM GhM ; see Eq. (8) in Section 2.2.), of the comx Oy g x Oy i peting amorphous and crystalline oxide overgrowths (see Fig. 5) on the bare (a) Mg{0001}, (b) Ti{0001} and (c) Si{111} substrates as function of oxide-film thickness (hfMx Oy g ) at a growth temperature of T0 = 298 K [3].


ð9bÞ

In practice, the unequivocal determination of solid surface energies from experimental quantities (e. g. fracture and cleavage energies) is extremely difficult, because generally a combination of surface energy and surface stress contributions is measured [60 – 65]. For a single-component system, the (excess) surface stress tensor, gSij , (which corresponds to the reversible work required to produce unit area of new surface by elastical stretching) is related to its surface energy, cS (i. e. the excess Gibbs energy per unit surface area, as defined by Eq. (9a)), according to [60, 64 – 67]: ! S qc gSij ¼ dij c S þ ð10Þ qeSij where dij is the Kronecker delta and eSij the elastic surface (excess) strain tensor. As follows from Eq. (10), the difficulty in 2 Eq. (9a) considers a solid phase A with a homogeneous (bulk) composition up to its surface with the ambient and, consequently, any compositional variations at its outer surface by e.g. surface segregation effects are not accounted for. Such compositional effects can be C P accounted for by adding an additional term, lj C j , to Eq. (9a),

j¼1

where C denotes the total number of components in the system and the term Cj corresponds to the surficial excess of component j per unit interface area (with a corresponding chemical potential, lj) (cf. Refs. [11, 67]).

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F


distinguishing between surface energies and surface stresses does not occur for liquids, because the diffusion of atoms in the liquid phase is fast enough to remove elastic strain contributions to the surface energy: i. e. qcS =qeSij ¼ 0 and thus gSij ¼ dij cS . In this case (i. e. for qcS =qeSij ¼ 0), the notion surface energy, cS , is often substituted by the notion surface tension, r S ð¼ c S ) [60, 64 – 67]; more precisely expressed: the numerical values of r S (e. g. in N m – 2) and c S (e. g. in J m – 2) then are the same. Besides the aforementioned surface stress contributions, the surface energy is also altered by its chemical interaction (i. e. equilibration) with the ambient (e. g., vacuum, a gas atmosphere, liquid phase, or an adsorbed layer). Therefore surface cleanness and controlled ambient conditions are crucial factors for the accurate and reproducible experimental determination of liquid and solid surface energies [60 – 64, 66]. Most experimentally determined values of, in particular solid, surface energies are affected by significant experimental uncertainties and errors. It is much easier to experimentally assess accurate values for the surface energy, cAS;m , of the liquid at the melting point, Tm, as derived by e. g. sessile or vertical plate experiments (cf. Refs. [59, 63, 68, 69] and references therein). In the following (Sections 3.1.1. and 3.2.1.) it is therefore proposed to estimate the solid surface energies, cAS ðT < Tm Þ, of amorphous and polycrystalline (elemental and homogenous compound) surfaces by extrapolation from the surface energy of the corresponding liquid phase at its melting point (provided that an experimental value for cAS;m is available), according to

or semiconductor) at temperature T can be expressed by (see Eqs. (11a) and (9b)) S;m S S cfAg ðTÞ ¼ cfAg þ qcfAg =qT ðT Tm Þ S;m S ffi cfAg SfAg =OfAg ðT Tm Þ

ð12Þ

S;m is the surface energy of the corresponding liquid where cfAg phase at its melting point, Tm. S;m have Comprehensive experimental data sets for cfAg been reported in, e. g., Refs. [58, 59, 63, 68, 70]. The molar S now is defined as the surface area, OfAg , in Eq. (12) (SfAg excess entropy of the system for one mole atoms {A} in the surface) corresponds to the total contact area with the ambient for one mole atomic (i. e. Wigner–Seitz) cells of S;m S , SfAg , and {A} surface atoms. Since the values of cfAg OfAg in Eq. (12) are intrinsically isotropic, the molar surface area OfAg can be directly related (on the basis of the macroscopic atom approach [44, 46]) to the molar volume, VfAg , of {A} at temperature T by: 2 ð13Þ OfAg ¼ ffAg C0 VfAg 3

The needed value for qcAS =qT upon application of Eq. (11a) or Eq. (11b), can be taken as a constant (cf. Refs. [44, 58, 59]), the value of which can either be derived from available experimental data or estimated on the basis of the macroscopic atom approach. Details are provided in the following Sections 3.1.1. and 3.2.1.

where ffAg represents the average fraction of the surface area of each atomic (i. e. Wigner–Seitz) cell in contact with the ambient (here: vacuum) and the proportionality constant C0 relates the surface area of one mole atomic (i. e. Wigner–Seitz) cells of the solid to its bulk volume (i. e. the 2=3 term C0 VfAg equals the sum of areas of one mole of atomic cells of A). Assuming a shape of the Wigner–Seitz cell of the {A} atoms in between a cube (i. e. ffAg ¼ 16) and a sphere (i. e. ffAg ¼ 12), it follows that [44, 46] the fraction ffAg 13 and the proportionality constant C0 = 4.5 · 108 mol – 1/3. The surface entropy for amorphous metals (and semiconS ductors) roughly equals SfAg *7.34 J mol – 1 K – 1 [43] S (note: the value of SfAg can be taken independent of the temperature and nearly equals the molar gas constant R = 8.3143 J mol – 1 K – 1 [44, 58]). Thus system-specific S S estimates for qcfAg =qT ffi SfAg OfAg can be obtained S using SfAg * 7.34 J mol – 1 K – 1 and by adopting a value of OfAg as calculated according to Eq. (13). Alternatively, S =qT some system-specific, experimental values for qcfAg can be taken from Refs. [58, 59, 63, 68]. The negative temS =qT ffi perature dependence of the surface energy (i. e. qcfAg S SfAg =OfAg < 0) is approximately the same for the liquid and the corresponding solid amorphous phase [44, 58, 59]. A S S =qT ð qchAi =qTÞ is: rough empirical estimate for qcfAg – 4 2 – 1 0:6 – 1:5 · 10 J m K [44, 58, 59] (see also Section 4.2.2.). Thus by adopting estimated or experimental values of S qcfAg =qT (see above), straightforward application of Eq. (12) S ðTÞ, of solid is possible to determine the surface energy, cfAg amorphous (semi-)metals at any given T from experimental S;m , of phase {A} at its melting values of the surface energy, cfAg point.

3.1. Surface energies of (semi-)metals

3.1.2. Crystalline (semi-)metal surfaces

3.1.1. Amorphous (semi-)metal surfaces

S The (orientation-specific) surface energy, chAi , (per unit area) of a crystalline solid, hAi, at temperature T, can be expressed by (see Eqs. (11b) and (9b))

cAS ðTÞ ¼ cAS;m þ qcAS =qT ðT Tm Þ

ð11aÞ

Experimental values for surface-orientation-specific energies of crystalline solid surfaces have only very scarcely been reported in the literature (especially for single-crystalline compound phases). Therefore, if such orientation-specific surface energies are required (e. g. for single-crystalline metal or oxide microstructures such as oxides; see Section 3.2.2.), but unavailable from the literature, estimated values may be obtained departing from theoretical values for such surface orientation-specific energies at 0 K, cAS;0 , as derived for “relaxed” (i. e. reconstructed) crystallographic surfaces at 0 K (by first-principle or molecular dynamics calculations). Next, extrapolation to the temperature of investigation can be performed according to (analogously to Eq. (11a)) cAS ðTÞ ¼ cAS;0 þ qcAS =qT T

ð11bÞ

S , (per unit area) of an undercooled The surface energy, cfAg (i. e. configurationally frozen) (semi-)metallic liquid phase, {A}, (as a model for the solid amorphous phase of the metal

1288

S;0 S;0 S S S chAi ðTÞ ffi chAi þ qchAi =qT T ffi chAi T ShAi =OhAi

ð14Þ


F

Feature


where the quantities in this equation pertaining to the solid crystalline phase hAi are defined analogously to those for S;0 the solid amorphous phase {A} in Eq. (12). Values of chAi and OhAi in Eq. (14) are generally anisotropic for crystalline solid surfaces (i. e. dependent on the crystallographic orientation of the considered surface plane), whereas the surface S , is approximately independent of the crystal entropy, ShAi surface orientation (and the temperature; cf. Eq. (11b)) [44, 59]. As discussed with respect to Eq. (11b) above, if reliable S ðTÞ are not availcrystallographic-specific values for chAi able from experiment, theoretical estimates for the orientaS;0 , may be applied. tion-specific surface energies at 0 K, chAi S;0 for the low-index crystalloSuch theoretical values of chAi graphic surfaces of many elements can be obtained from e. g. Refs. [69, 71] and references therein. The subsequent extrapolation to the temperature of investigation according to Eq. (14) can be performed employing estimated or experimental values for qcShAi =qT as follows. Some experimental, “average-crystal-plane” values of S =qT for certain crystalline metals (and semi-conductors) qchAi have been reported in e. g. Refs. [59, 63, 68] S S =qTð qcfAg =qTÞ 1:5 ± 0:6 · 10 – 4 J m2 K – 1). Alter(qchAi S =qT ffi natively, a “crystal-plane-specific” estimate of qchAi S S ShAi =OhAi can be obtained by using ShAi *7.72 J mol – 1 K – 1 [43] in Eq. (14) and by adopting an orientation-specific value of OhAi , as approximated by the projected area enclosed by one mole of hAi atoms allocated to a corresponding crystallographic plane of hAi parallel to the surface [2, 3, 41]. S It follows that values of chAi ðTÞ are lower for more densely packed crystallographic surfaces (e. g. the non-reconstructed {111}, {110}, and {0001} surface planes for fcc, bcc, and hcp metals, respectively), as well as that S S > cfAg [44, 58, 59, 69, 71]. chAi Alternatively, for polycrystalline solid surfaces, an S;0 “average-crystal-plane” value for chAi ðTÞ can be obtained by extrapolation from the corresponding experimental vaS;m , after its multiplication with lue at the melting point, cfAg an empirical correction factor of 1.13 (recognizing the higher density of the crystalline phase, i. e. VhAi < VfAg and that fhAi > ffAg [59]). A corresponding “average-crysS S =qT ffi ShAi OhAi can be obtal-plane” estimate for qchAi S *7.72 J mol – 1 K – 1 (see above) in combitained using ShAi nation with an “average-crystal-plane” value for OhAi taken as (cf. Eq. (13)): 2

OhAi ¼ fhAi C0 ½VhAi 3

ð15Þ

with fhAi = 0.35 [44, 59] and C0 = 4.5 · 108 mol – 1/3.

S;m Fig. 7. Surface energy, cfM , of liquid oxides at their melting point, Tm , x Oy g m =ðNA xÞ2=3 . versus the corresponding energy term, kB Tm ½VfM x Oy g The dashed line represents a linear fit through the data points according to Eq. (16) in Section 3.2.1. [3, 74].

S;m cfM can be obtained from an established empirical relax Oy g S;m m and the molar volume, VfM , tionship between cfM x Oy g x Oy g of the oxide phase at Tm (Fig. 7) [74]: V m 23 fMx Oy g S;m cfMx Oy g ffi 1:764 kB Tm 0:0372 ðJ m2 Þ ð16Þ NA x

where x is the number of metal ions per Mx Oy unit “molecule” (kB and NA denote Boltzman’s constant and Avogadro’s constant, respectively). S , of molten oxides typically The surface energies, cfM x Oy g have only a very weak negative temperature dependence S;m with an averaged value of about qcfM =qT* – 0.7 ± 0.5 · x Oy g –4 2 –1 10 J m K [3] (which is lower than the corresponding empirical estimate for metals and semiconductors of S S =qT qcfAg =qT* – 1.5 ± 0.6 · 10 – 4 J m2 K – 1; see qchAi Sections 3.1.1. and 3.1.2.). Otherwise, a rough estimate for S S =qT ffi SfM =OfMx Oy g ðTÞ can be obtained using qcfM x Oy g x Oy g –1 S SfMx Oy g *7.34 J mol K – 1 (see Section 3.2.1.). Finally, it is noted that, only for some (molten) networkforming oxides (e. g. GeO2, B2O3, and V2O5), surprisingly, S has a very weak positive temperature dependence of cfM x Oy g S been found: qcfM =qT*+ 0.4 ± 0.3 · 10 – 4 J m – 2 K – 1 (see x Oy g references listed in Ref. [74]).

3.2. Surface energies of oxides

3.2.2. Crystalline oxide surfaces

3.2.1. Amorphous oxide surfaces

Experimental values for the orientation-specific surface energies of crystalline oxides are extremely scarce (only literature values for clean MgO{100} surfaces were found [75, 76]). Therefore (as for the single-crystalline (semi-)metal surfaces; see Section 3.1.2.), surface-orientation-specific esS ðTÞ are generally determined by extrapolatimates of chM x Oy i tion (see Eq. (11b)) from corresponding theoretical values for cS;0 hMx Oy i at 0 K, as calculated for orientation-specific oxide surface planes of minimized energy (i. e. for “relaxed” oxide surface terminations of lowest energy) by first principle or molecular dynamics simulation methods (cf. tabulated val-

S , of solid amorphous oxEstimated surface energies, cfM x Oy g ides can be obtained in the same way as proposed here for amorphous metals and semiconductors (see Section 3.1.1. and Eq. (11a)): i. e. by extrapolation from the surface enS;m , of the liquid oxide at its melting point, Tm, ergy, cfM x Oy g using an experimental or estimated value for the temperaS =qT (see below). ture dependence, qcfM x Oy g S;m have been reported Some experimental values of cfM x Oy g in, e. g., Refs. [70, 72, 73]. Alternatively, an estimate of


1289

F


S;0 ues in Ref. [3]). Alternatively, a rough estimate of chM for x Oy i any low-index crystalline oxide surface can be obtained from an established empirical relationship [3]: S;0 chM x Oy i

ffi 0:0105

NN NMO

lattice EhM x Oy i

y

1

ð17Þ

NN denotes the molar number of broken, nearestwhere NMO neighbour bonds at the oxide surface per unit surface area (which depends on the crystallographic surface plane conlattice is the lattice energy (i. e. the Gibbs energy sidered), EhM x Oy i to form the oxide from its respective ions at T = 0 K) and y is the number of oxygen ions per Mx Oy molecule. An (even) S;0 has been more accurate empirical relationship for cfM x Oy g established for the {100}, {110} and {111} crystallographic faces of the oxide phases with a rock salt structure (Fig. 8) [3], i. e. S;0 lattice 0 ffi EhMOi VhMOi chMOfhklgi

23

NA

13

ð18Þ

with U = 0.012, U = 0.026, and U = 0.028 for the {100}, 0 {110}, and {111} crystallographic faces, respectively (VhMOi denotes the molar volume of the oxide at T = 0 K). Furthermore, the surface energy (at T = 0 K) of high-index oxide surfaces can be approximated using a “surface step” model (i. e. by assuming that the high-index oxide surface consists of stepped terraces of the corresponding low-index surfaces) [77]. The “average-crystal-plane” temperature dependence, S;m qchM =qT, can be taken equal to that for the correx Oy i sponding amorphous oxide phase (Section 3.2.2.), or else a rough “orientation-specific” estimate can be obtained S S from qcShMx Oy i =qT ffi ShM =OhMx Oy i with ShM * x Oy i x Oy i 7.72 J mol – 1 K – 1 (see Section 3.2.1.).

face per unit interface area [67]: cAjB ðTÞ ¼

HAjB T SAjB OAjB

ð19Þ

where HAjB ð¼ HAþB HA HB Þ and SAjB ð¼ SAþB SA SB Þ are the excess enthalpy and excess entropy of the system due to the presence of the AjB interface area, OAjB : i. e. the differences between the actual total enthalpy (HAþB ) and total entropy (SAþB ) of the system and the sum of the enthalpies (i. e. HA þ HB ) and entropies (SA þ SB ) of the individual homogenous solids A and B in the absence of the interface, i. e. if they were undisturbed by the dividing AjB interface [67]. Compositional variations of the bulk solids A and B in the vicinity of the adjoining AjB interface by e. g. interfacial segregation effects are not accounted for in Eq. (19) (see Footnote 2 and, e. g., Refs. [11, 67]). Direct, unequivocal quantitative determination of the energy of an interface between two solids (denoted as SS interface) is not possible by experiment (see what follows). Indirect determination of the SS interface energy, cAjB , is attempted by experiment by relating measured quantities (e. g. adhesive forces, interfacial fracture, cleavage energies, groove shape) to fundamental quantities, such as the work of adhesion, fracture and/or separation [78 – 83]. One such fundamental quantity, the ideal work of separation, sep , is defined as the reversible work, performed in a kind WAjB of “Gedankenexperiment”, to separate the system at the solid–solid AjB interface, thereby creating two free surfaces of the corresponding solids A and B, whereby plastic and diffusional degrees of freedom upon separation are supsep is expressed posed to be suppressed [84]. The value of WAjB by the ideal Dupré equation, i. e. sep ¼ cAS;unrelaxed þ cBS;unrelaxed cAjB WAjB

ð20Þ

4. Assessment of solid–solid interface energies

where cAS;unrelaxed and cBS;unrelaxed denote the energies of the respective “unrelaxed” A and B surfaces at infinite separation (i. e. the instantaneous values after cleavage; before equilibration of the fresh surfaces with the ambient). Unfortunately, any attempt to determine (indirectly) a value for a SS interface energy, cAjB , from the measured strength of the SS interface is affected by numerous (also interdependent) factors and side-effects, such as the geometry of the loading, the plastic and elastic properties of A and B, residual internal strains, defect formation and dislocation movement (i. e. plastic flow), crack formation and propagation, the presence and size of flaws (e. g. interface roughness, chemical impurities), diffusional processes for chemical equilibration (e. g. chemical interaction of the free surfaces with the ambient; surface segregation) [78, 80, 82 – 84]. Therefore, the energy consumed in any conceivable cleavage experiment generally substantially deviates sep accordfrom (i. e. exceeds) the fundamental value of WAjB ing to Eq. (20) [78, 84, 85], thereby invalidating reliable determination of cAjB (supposing that accurate values for cAS;unrelaxed and cBS;unrelaxed are available3).

In analogy with the definition of the solid surface energy (Section 3.1.), the energy, cAjB , per unit area of the interface between two homogeneous solid phases A and B at temperature T can be defined as the excess Gibbs energy of the atoms or molecules of A and B associated with the inter-

3 Note that, for a crystalline solid of e.g. phase hAi, it holds that the “unS; unrelaxed , representing the created solid surrelaxed” surface energy, chAi face of hAi before its equilibration with the ambient, is different from (i. e. generally higher than) the respective “relaxed” surface energy, S , which corresponds to the solid surface of hAi in equilibrium with chAi the ambient.

S;0 Fig. 8. Surface energies, chMOi , of the low-index crystallographic faces of crystalline oxides with a rock-salt structure at T = 0 K versus the enlattice 0 2=3 ergy term, EhMOi VhMOi NA1=3 . The indicated value of U corresponds to the slope of the line fitted through the concerned data points according to Eq. (18) in Section 3.2.2. [3].

1290


F

Feature


On the contrary, it is much easier to obtain accurate experimental measures for the energy, chAijfBg , per unit area of the interface between a crystalline solid phase, hAi, and a liquid phase {B}, from the measured contact angle, h, of a liquid drop of phase {B} on the surface of the crystalline solid hAi, by application of Young’s equation cos h ¼

S chAi chAijfBg S cfBg

ð21Þ

S S where chAi and cfBg denote the surfaces energies of crystalline hAi and liquid {B} in equilibrium with the ambient. Thus obtained experimental values for solid–liquid (SL) hAij{B} interface energies, chAijfBg , are often considered as approximate measures for the corresponding solid–solid hAijhBi interface energies, chAijhBi . However, the energy of the interface between two crystalline solids hAi and hBi generally comprises excess energy contributions due to the mismatch between the adjoining crystalline lattices of hAi and hBi at the hAijhBi interface (i. e. homogeneous residual strain and inhomogeneous residual strain induced by misfit dislocations; see Section 4.3.). Such structural energy contributions due to the lattice mismatch between adjoining crystalline solids do not occur for the corresponding solid–liquid hAij{B} interface (Section 4.2.) and therefore it often holds that chAijfBg < chAijhBi [1, 2, 41] (see also Sections 1, 4, 5 and 6). Thus, up to date, reliable values for the interface energy between two crystalline solids can (still) only be assessed by theoretical approaches either on the basis of first principle calculations (e. g. by density functional theory; DFT) or by application of semi-empirical calculation methods (e. g. tight-binding method, molecular orbital theory) or by application of (semi-)empirical formulations as derived from experimental data sets: cf. Refs. [3, 21, 22, 38, 84 – 90] and references therein. Unfortunately, first-principle calculations, but also the mentioned (semi-)empirical formulations, of SS interface energies rely on detailed preknowledge of the interface structure (i. e. the precise coordinates and types of atoms at or near the interface), which can only be very elaborately determined experimentally by quantitative high-resolution transmission electron microscopy (QHRTEM), spatially-resolved electron energy-loss spectroscopy (EELS), and/or spatially-resolved electron energy-loss near-edge structure spectroscopy (ELNES) (in combination with delicate sample preparation methods) [22, 84, 91, 92]. It is concluded that, up to date, versatilely applicable descriptions of solid–solid interfacial energies as a function of, e. g., phase composition and structure, temperature and crystallographic orientation of the interface, which knowledge is mandatory for the thermodynamic treatment presented in this paper, can only be readily and successfully assessed by employing semi-empirical expressions, in particular those derived on the basis of the macroscopic atom approach [1 – 3, 40 – 46], as dealt with in the following sections.

moderate bond flexibility [2, 53, 54, 93, 94]. Therefore, upon creation of an interface between two amorphous phases, {A} and {B}, joining opposing (i. e. positive) energy contributions to the resultant interface energy, cfAgjfBg , due to mismatch strain between {A} and {B} at the {A}j{B} interface can be ignored. The resultant {A}j{B} interface energy, cfAgjfBg , then only contains excess (cf. begin of Section 4) enthalpy and entropy energy contributions due to the physical (i. e. Van der Waals) and chemical (i. e. metallic, covalent and/or ionic (i. e. electrostatic)) interactions of {A} and {B} across the {A}j{B} interface [44, 84, 88, 95, 96]: HfAgjfBg T SfAgjfBg entropy interaction cfAgjfBg ¼ ¼ cfAgjfBg þ cfAgjfBg ð22Þ OfAgjfBg interaction The interface enthalpy energy contribution, cfAgjfBg ¼ interaction =OfAgjfBg (per unit area interface), due to the inHfAgjfBg teractions between two adjoining amorphous phases {B} and {A} across the {A}j{B} interface is given by [2, 43 – 45] (see also Eq. (13)): " # 1 1 1 ffAg D HfAg!fBg ffBg D HfBg!fAg interaction cfAgjfBg ¼ þ 2 OfAg OfBg

ffi

1 1 D HfAg!fBg þ D HfBg!fAg 2=3

2=3

C0 ðVfAg þ VfBg Þ

ð23aÞ

4.1. Amorphous–amorphous interfaces between (semi-)metals

1 1 and D HfBg!fAg denote the partial enwhere D HfAg!fBg thalpies of dissolving one mole {A} in {B} and of one mole {B} in {A}, at infinite dissolution, respectively; OfAg and OfBg correspond to the molar interface areas of {A} and {B} (see Eq. (13) in Section 3.1.1.). 1 1 Values for D HfAg!fBg and D HfBg!fAg at the temperature concerned (and typically at 1 atm pressure) can easily be extracted from corresponding phase diagrams, e. g. using the Thermo-Calc software package [51]. Otherwise, such values, as determined experimentally, can be found in Refs. [44, 49, 50]. A note about the definition of a state of reference for the interface energy calculations is in order. It appears natural 1 1 and D HfBg!fAg in Eq. (23a) with reto define D HfAg!fBg spect to A and B in the (undercooled) liquid state. However, a direct comparison between calculated values for amorphous–amorphous, crystalline–amorphous and crystalline– crystalline SS AjB interface energies (see Sections 4.1., 4.2.1. and 4.3.1., respectively) is only possible if all partial enthalpies are defined with respect to the same reference states for the components. In the following, hAi and hBi in their most stable crystalline modification at a temperature of 298 K and a pressure of 1 atm are chosen as reference states for the employed partial enthalpies, as designated by an addi1;cr 1;cr and D HfBg!fAg . As tional superscript, “cr”: i. e. D HfAg!fBg a result, the expression for the interface enthalpy energy contribution for the {A}j{B} interface between amorphous {A} and amorphous {B} becomes (cf. Eq. (23a)) 1;cr 1;cr D HfAg!fBg þ D HfBg!fAg interaction cfAgjfBg ffi ð23bÞ 2=3 2=3 C0 ðVfAg þ VfBg Þ

Viscous flow in amorphous phases (considered as undercooled liquids) is relatively easy (as compared to crystalline solids), because of their relatively large free volume and

1;cr 1 ffi D HfAg!fBg þ D HhBi!fBg where D HfAg!fBg 1;cr 1 and D HfBg!fAg ffi D HfBg!fAg þ D HhAi!fAg with D HhBi!fBg ¼ ½D HfBg ðTÞ D HhBi ðTÞ and D HhAi!fAg ¼


1291

F


½D HfAg ðTÞ D HhAi ðTÞ (at the temperature concerned and at 1 atm pressure), respectively. For comprehensive discussion on the necessity of explicit definition of the thermodynamic reference state, see Refs. [43, 44, 46]. entropy The interface entropy contribution, cfAgjfBg ¼ T SfAgjfBg OfAgjfBg (per unit area interface) predominantly originates from the change in vibrational entropy of the interacting {A} and {B} phases, as compared to the “bulk” of these amorphous phases, at the {A}j{B} interface. Assuming similar sizes of the atomic cells of the interacting {A} and {B} atoms at the {A}j{B} interface (i. e. 2=3 2=3 entropy for the {A}j{B} inVfAg VfBg ), an estimate of cfAgjfBg terface between two amorphous (or liquid) metals {A} and {B} is obtained from [43, 97]:4 1 T ffAg 3R centropy fAgjfBg ¼ 1 O fAg þ OfBg 2 qffiffiffiffiffiffiffiffiffiffi qffiffiffiffiffiffiffiffiffiffi 2 3 HfAg HfBg 5 ln41 H þ H þ DH fAg fBg fAgjfBg 2

ffi

6R T 2=3

2=3

C0 VfAg þ VfBg

qffiffiffiffiffiffiffiffiffiffi qffiffiffiffiffiffiffiffiffiffi 3 HfAg HfBg 5 ln41 H þ H þ DH fAg fBg fAgjfBg 2 2

ð24aÞ

Here HfAg and HfBg denote the Debye temperatures of {A} and {B}; DHfAgjfBg corresponds to the average change of the Debye temperature of the {A} and {B} interface atoms with respect to the average Debye temperature of the corresponding bulk phases of {A} and {B}, which can be expressed by [43, 97] DHfAgjfBg ¼ 34:1 10

3

ðDHfABg =RÞ

ð24bÞ

where DHfABg ðTÞ denotes the temperature-dependent (bulk) enthalpy of formation of the {AB} solid solution of equiatomic composition. Substitution of Eqs. (23), (24a) and (24b) in Eq. (22) finally leads to Eq. (25) (see bottom of page). 4.2. Crystalline–amorphous interfaces With reference to the discussion of the interface between two amorphous solids (Section 4.1.), it can be assumed that mismatch strain is also absent for the interface between a crystalline phase hAi and an amorphous phase, {B}. The re4 Similar (approximate) expressions for the interfacial vibrational entropy contribution can be derived for {A}|hABi or hAi|{AB}interfaces on the basis of the guidelines provided by Ref. [43].

2 1;cr 1;cr D HfAg!fBg þ D HfBg!fAg 6 R T ln41

cfAgjfBg ¼

1292

sultant hAij{B} interface energy, chAijfBg , can then be expressed as the resultant of three additive interfacial energy contributions [1 – 3, 43 – 45]: chAijfBg ¼

HhAijfBg T ShAijfBg OhAijfBg

enthalpy entropy interaction ¼ chAijfBg þ chAijfBg þ chAijfBg

enthalpy The enthalpy contribution, chAijfBg , arises from the relative increase in enthalpy of crystalline hAi at the hAij{B} interface (as compared to bulk crystalline hAi) due to the liquid-type of bonding of hAi with amorphous {B} at the hAij{B} interface [1, 2, 43 – 45, 98]. The interaction coninteraction , results from the physical and chemical tribution, chAijfBg interactions of hAi and {B} across the hAij{B} interface (cf. Section 4.1.). Furthermore it is assumed that the vibrational entropy does not change in the hAi and {B} phases at the hAij{B} interface (cf. Eq. (24a) in Section 4.1.), but it is recognized that the configurational entropy of amorphous {B} is lowered at the hAij{B} interface due to an ordering effect imposed by the periodicity of the interacting crystalline hAi phase at the interface [98 – 100] (for experimental support of this phenomenom, see Ref. [101]).

4.2.1. Crystalline–amorphous interfaces between (semi-)metals For the interface between a solid crystalline metal, hAi, and a solid amorphous metal, {B}, the excess enthalpy enthalpy contribution, chAijfBg (cf. Eq. (26)), due to the relative increase in enthalpy of crystalline hAi at the interface (as compared to bulk crystalline hAi), is approximated by [1, 2, 43 – 45, 98] enthalpy chAijfBg ¼

fhAi D HhAi!fAg HfAg ðTÞ HhAi ðTÞ ffi 2=3 OhAi C0 V

ð27Þ

hAi

with fhAi % 0.35 (Section 3.1.2.). interaction The interfacial interaction contribution, chAijfBg , is given by (cf. Eq. (23) in Section 4.1. with fhii ffi ffig for i = A, B [44]) " # 1;cr 1;cr 1 fhAi D HfAg!fBg ffBg D HfBg!fAg interaction chAijfBg ¼ þ 2 OhAi OfBg 2 3 1;cr 1;cr 1 4D HfAg!fBg D HfBg!fAg 5 ð28Þ þ ffi 2=3 2=3 2 C0 VhAi C0 VfBg 1;cr 1 ffi D HfAg!fBg þ D HhBi!fBg and with D HfAg!fBg 1;cr 1 D HfBg!fAg ffi D HfBg!fAg þ DHhAi!fAg (see Section 4.1.) and where it has been recognized that the layer of hAi adjacent to {B} can thermodynamically be approximated by {A} [1, 2, 43 – 45, 98].

qffiffiffiffiffiffiffiffiffiffi qffiffiffiffiffiffiffiffiffiffi HfAg HfBg

ðHfAg þ HfBg Þ þ 34:1 103 ðD HfABg =RÞ

2=3 2=3 C0 VfAg þ VfBg 2

ð26Þ

3 5 ð25Þ


F

Feature


entropy The interfacial entropy contribution, chAijfBg , due to the decrease in configurational entropy of the amorphous phase {B} at the hAij{B} interface, is estimated by [1, 99, 100]: entropy ¼ T chAijfBg

0:678 R 0:678 R T ffi 2=3 OfBg ffBg C0 VfBg

ð29Þ

with ffBg ffi 13 . Substitution of Eqs. (27 – 29) into Eq. (26) yields chAijfBg ffi

1;cr ½HfAg ðTÞ HhAi ðTÞ þ 12 D HfAg!fBg 2=3

C0 VhAi þ

1;cr 0:678 R T ð ffBg Þ1 þ 12 D HfBg!fAg 2=3

C0 VfBg

ð30Þ

4.2.2. Crystalline–amorphous interfaces between metals and oxides For the interface between a crystalline metal, hMi, and its amorphous oxide, fMx Oy g, the enthalpy contribution, enthalpy (cf. Eq. (26)), due to the enthalpy increase of chMijfM x Oy g crystalline hMi at the hMijfMx Oy g interface, with respect to the enthalpy of “bulk” hMi, due to the liquid-type of bonding with the amorphous oxide phase, is estimated by (cf. Eq. (17) in Section 4.2.1. and Ref. [2]) fhMi D HhMi!fMg fhMi HfMg ðTÞ HhMi ðTÞ enthalpy chMijfMx Oy g ¼ ¼ OhMi OhMi ð31Þ with fhMi % 0.35 (Section 3.1.2.). The molar interface area, OhMi , of metal atoms of hMi at the hMijfMx Oy g interface follows from the area enclosed by one mole of metal atoms in the crystallographic plane of hMi at the hMijfMx Oy g interface plane (alternatively, an “average-crystal-plane” val2=3 ue of OhMi is obtained from OhMi ffi fhMi C0 VhMi ; see Eq. (15) in Section 3.1.2.). interaction , due The interfacial interaction contribution, chMijfM x Oy g to the interactions of hMi and fMx Oy g across the hMijfMx Oy g interface, is given by (see Section 4.1. and Ref. [2]) interaction chMijfM ¼ x Oy g

1 ffOg D HO!hMi

OfOg

ð32Þ

1 where D HO!hMi denotes the partial enthalpy of dissolving one mole O atoms at infinite dissolution in (solid) crystalline hMi and OfOg is the interface area enclosed by one mole of oxygen ions in the amorphous oxide at the hMijfMx Oy g interface (ffOg ffi 13; see Section 3.1.1.). Evidently, for metal j oxide interfaces hM I ijfMxII Oy g where M I 6¼ M II , an additional (excess) interaction energy term, 1;cr ffM II g D HfM II g!fM I g =OfM II g , (cf. Eq. (28) and related dis-

chMijfMx Oy g ¼

cussion) should to be added to the interfacial interaction contribution according to Eq. (32), to account for the physical and chemical interactions between dissimilar metal atoms across the hM I ijfMxII Oy g interface: for details, see Refs. [2, 40, 102]. entropy The entropy contribution, chMijfM , due to the decrease x Oy g in configurational entropy of the amorphous phase fMx Oy g at the hMijfMx Oy g interface, can be estimated from the entropy difference between bulk amorphous fMx Oy g and the corresponding bulk crystalline oxide phase, hMxOyi, i. e. ShMx Oy i SfMx Oy g [2]. The entropy contribution per unit area interface then becomes: ShMx Oy i ðTÞ SfMx Oy g ðTÞ entropy chMijfM ¼ T ð33Þ x Oy g y OfOg where y equals the number of O ions per stoichiometric Mx Oy molecule (and OfOg has been defined below Eq. (32)). Substitution of Eqs. (31 – 33) in Eq. (26) gives Eq. (34) (see bottom of page). For those metal–oxide systems for which the value of 1 D HO!hMi is unknown, a value can be estimated from the 1 (in [J (mole established empirical relation between D HO!hMi –1 O) ]) and the corresponding enthalpy of formation, f D HhM (in [J (mole MxOy) – 1]) of the crystalline oxide, x Oy i hMxOyi, out of its stable elements [2, 3]: 1 f 5 ffi 1:2 y1 DHhM ð35Þ DHO!hMi O i ðTÞ þ 1 10 x

y

For most metal–oxide systems, the metal–oxygen bond 1 formation [and thus the value of DHO!hMi employed in Eq. (34)] is strongly exothermic [3, 103]. Consequently, the resultant crystalline–amorphous interface energy, chMijfMx Oy g , is generally predominated by the relatively large negative interaction metal–oxygen interaction energy, chMijfM [2, 3] (note: this x Oy g also holds for the corresponding crystalline–crystalline interface energy, chMijhMx Oy i [42]; see Section 4.3.2.). For enthalpy example, the positive sum of the enthalpy (chMijfM ) and x Oy g entropy (centropy ) contributions is typically smaller than hMijfMx Oy g + 0.3 ± 0.2 J m – 2, whereas the corresponding negative ininteraction , is in the range of teraction contribution, chMijfM x Oy g interaction interaction – 4.5 J m – 2 to –1.0 J m – 2 (e. g. chMgijfMgOg ffi chAlijfAl ffi 2 O3 g interaction chZrijfZrO *– 4.5 ± 0.2 J m – 2; 2g

interaction chSiijfSiO , 2g

interaction interaction chNiijfNiOg and chFeijfFe are in between – 2 and –1 J m – 2; 2 O3 g interaction as an exception chCuijfCuO *– 0.2 ± 0.1 J m – 2) [2, 3, 41, 2g 104]. This implies that the lowest value of chMijfMx Oy g , and thus the most stable (i. e. thermodynamically-preferred) hMijfMx Oy g interface, is generally achieved by maximizing the density of metal–oxygen bonds across the hMijfMx Oy g interface (i. e. the number of metal–oxygen bonds per unit interface area), which results in a (random) dense packing of the amorphous oxide at the hMijfMx Oy g interface, in accordance with recent experimental observations [42]. A good approximate for the molar interface area, OfOg , enclosed by one mole of oxygen atoms in the amorphous oxide at the hMijfMx Oy g interface (see Eq. (34)), is

1 1 fhMi ½HfMg ðTÞ HhMi ðTÞ ffOg D HO!hMi T y ½ShMx Oy i ðTÞ SfMx Oy g ðTÞ þ OfOg OhMi


interaction chCrijfCr , 2 O3 g

ð34Þ 1293

F


therefore obtained from such area enclosed by one mole of O ions in the most densely-packed plane of the corresponding (unstrained) crystalline oxide phase, hMxOyi, parallel to the interface (independent of the adjacent crystallographic face of the metal substrate hMi) [2]. 4.3. Crystalline–crystalline solid interfaces The interface energy, chAijhBi , of the hAijhBi interface between two crystalline solid compounds hAi and hBi is the resultant of a chemical and a structural term [2, 41, 44, 45, 105]. The chemical term accounts for the enthalpy and entropy contributions due to the physical and chemical interactions between hAi and hBi across the interface (as already introduced in Sections 4.1. and 4.2.), whereas the structural mismatch ) originates from the term (further designated as chAijhBi mismatch between the adjoining crystal structures of hAi and hBi at the hAijhBi interface. For a fully coherent interface, all lattice mismatch is accommodated elastically by hAi and/or hBi at the hAijhBi boundary plane. This limiting case, which will be further referred to as the “elastic regime”, generally only occurs for small initial lattice mismatches at the hAijhBi boundary plane of, say, up to 5 %, dependent on, e. g., the A–B bond strength, the mechanical properties and the individual (layer) thicknesses of hAi and/or hBi (see, e. g., Refs. [41, 67] and references therein). More commonly, the initial mismatch strains in hAi and/or hBi are largely relaxed by built-in misfit dislocations at the hAijhBi interface. For this intermediate case (further referred to as the “mixed regime”), the residual homogeneous strain , can thought to be superstrains in hAi and/or hBi, chAijhBi imposed on the periodic, inhomogeneous strain field, dislocation , resulting from the sum of strain fields associated chAijhBi with each of the misfit dislocations at the semi-coherent hAijhBi interface [41, 106], i. e.

chMijhMx Oy i , is generally dominant for hMijhMx Oy i interfaces and, consequently, the density of metal–oxygen bonds across metaljoxide interfaces, as determined by the orientations of the adjoining crystallographic planes of the metal and the oxide at the hMijhMx Oy i interface (see e. g. Refs. [3, 41, 42, 56] and Section 4.3.2.), often plays a crucial role for the interface thermodynamics of oxide phases in contact with metals [42, 56]. On the contrary, the crystallographic orientation relationship (OR) between adjoining crystalline (semi-)metals can often, to a first approximation, be disregarded in thermodynamic model descriptions of phase transformations in (small-dimensional) (semi-)metal systems (i. e. average-crystal plane values can be adopted, instead of orientation-dependent values for the molar interface areas; cf. Section 3.1.1.) [1, 4, 17, 43 – 45], because the corresponding crystalline– crystalline interface energies are generally not dominated by the interaction energy contributions, as also holds for the amorphous–crystalline interface energies (see Sections 4.3.1. and 4.2.1., respectively). 4.3.1. Crystalline–crystalline interfaces between (semi-)metals For the hAijhBi interface between a solid crystalline metal, hAi, and a solid crystalline metal, hBi, the interfacial interinteraction action (chAijhBi ) and entropy (centropy hAijhBi ) contributions in Eq. (36) are given by (cf. similar expressions for the {A}j{B} interface in Sections 4.2.1. and 4.1.) " # 1;cr 1;cr f D H f D H 1 hAi hBi hAi!hBi hBi!hAi interaction ¼ þ chAijhBi 2 OhAi OhBi

entropy interaction mismatch þ chAijhBi þ chAijhBi chAijhBi ¼ chAijhBi entropy interaction strain dislocation ¼ chAijhBi þ chAijfBg þ chAijhBi þ chAijhBi

ð36Þ

If all residual mismatch strains in hAi and hBi are fully relaxed by the generation of misfit dislocations at the hAijhBi interface, the (fully) “plastic regime” has been entered strain mismatch dislocation ¼ 0 and chAijhBi ¼ chAijhBi ). (i. e. chAijhBi For the interface energies between two crystalline solids, as a rule of thumb, the interfacial interaction energy contribuinteraction , to the resultant interface energy, is much lartion, chAijhBi ger (i. e. less negative or even slightly positive) for interfaces interaction values in the range of between (semi-)metals (with chAijhBi –2 –2 – 0.5 J m to +1.0 J m ) than the similar contribution for interfaces between (semi-)metals and oxides (with negative interaction values in the range of about – 4.5 J m – 2 to chMijhM x Oy i – 1.0 J m – 2; cf. Section 4.2.2.). Similar to hMijfMx Oy g interfaces (see Section 4.2.2.), the interaction to the resultant interface energy, contribution of chMijhM x Oy i 2 1;cr 1;cr D HhAi!hBi þ D HhBi!hAi 6R T ln41

chAijhBi ¼

1294

1;cr 1;cr D HhAi!hBi þ D HhBi!hAi

ffi 2=3 2=3 C0 VhAi þ VhBi

ð37Þ

6R T

2=3 2=3 C0 VhAi þ VhBi

ð38Þ

and entropy chAijhBi ffi

qffiffiffiffiffiffiffiffiffi qffiffiffiffiffiffiffiffiffi 3 HhAi HhBi

5 ln4 1 3 D H H þ H =R þ 34:1 10 hAi hBi 2 hABi 2

where D HhABi is defined as the enthalpy of formation of the 1;cr solid solution of equiatomic composition; DHhAi!hBi 1;cr and DHhBi!hAi denote the partial enthalpies of dissolving one mole hAi in hBi and of one mole hBi in hAi at infinite dissolution, respectively (with the crystalline components hAi and hBi taken as the reference state; see Section 4.1.). The lattice mismatch contribution can be related to the averaged energy of a large-angle grain boundary in the corqffiffiffiffiffiffiffiffiffi qffiffiffiffiffiffiffiffiffi HhAi HhBi

3

5 S S ðHhAi þ HhBi Þ þ 34:1 103 ðD HhABi =RÞ chAi ðTÞ þ chBi ðTÞ

þ 2=3 2=3 6 C0 VhAi þ VhBi 2

ð40Þ


F

Feature


responding crystalline metals, which empirically equals [45, 107] about one third of the surface energy of the respective crystalline metal in contact with vacuum (as defined in Section 3.1.2.), i. e. " S # S S S c ðTÞ þ c ðTÞ chAi ðTÞ þ chBi ðTÞ 1 hAi hBi mismatch ¼ ð39Þ chAijhBi ¼ 3 2 6 Substitution of Eqs. (37 – 39) in Eq. (36) gives Eq. (40) (see bottom of previous page). 4.3.2. Crystalline–crystalline interfaces between metals and oxides

i

For (semi-)coherent hMijhMx Oy i interfaces between a crystalline metal, hMi, and its crystalline oxide, hMxOyi, the excess entropy contribution to chMijhMx Oy i is negligibly small interaction mismatch (as compared to chMijhM and chMijhM ; see introductory x Oy i x Oy i part of Section 4.3.) and is therefore neglected. In the following, the interfacial interaction, strain, and dislocation contributions to chMijhMx Oy i (see Eq. (36)) will be evaluated for the case of an overgrowth of an ultra-thin oxide film (of, say, hhMx Oy i < 10 nm) on its bare single-crystalline metal surface (cf. Section 2.2.), according to the basic concept outlined in Refs. [2, 41]. For the limiting case of an initial epitaxial overgrowth of hMxOyi on hMi (i. e. in the fully elastic regime; see introduction of Section 4.3.), all lattice mismatch is accommodated fully elastically by the thin epitaxial oxide film: i. e. dislocation = 0. The homogeneous, normal strains, e11 and chMijhM x Oy i e22, in the oxide film in mutually perpendicular directions 1 and 2 along the coherent hMijhMx Oy i interface plane, are then given by the initial lattice mismatch values f1 and f2 along the corresponding directions 1 and 2 within the interface plane, respectively (as governed by the OR between hMi and hMxOyi): i

eii ¼ fi ¼

ahMi i ahMx Oy i ia hMx Oy i

ði ¼ 1; 2Þ

ð41Þ

where i ahMi and i ahMx Oy i denote values of unstrained lattice spacings of hMi and hMxOyi along the corresponding perpendicular directions 1 and 2, respectively, within the interface plane. However, with increasing oxide film thickness, hhMx Oy i , as well as at the initial stage of crystalline oxide formation for hMijhMx Oy i systems of large initial lattice mismatch (larger than, say, *5 %), any mismatch/growth strain in the crystalline oxide film generally has been partly or fully relaxed by built-in misfit dislocations at the hMijhMx Oy i interface. With increasing density of misfit dislocations strain , deat the interface, the strain contribution, chMijhM x Oy i dislocation creases, whereas the dislocation contribution, chMijhM , x Oy i increases [41, 106]. In the above sketched situation (i. e. in the mixed regime), at a given oxide thickness, hhMx Oy i , and growth temperature, interaction , T, the interfacial interaction energy contribution, chMijhM x Oy i then is given by [41] (cf. Eq. (32) in Section 4.2.2.) interaction chMijhM ¼ x Oy i

1 fhOi D HO!hMi

OhOi

ð1 þ e11 Þ ð1 þ e22 Þ

where the molar interface area, OhOi , pertains to the “hypothetically” unstrained crystal plane of hMxOyi at the hMijhMx Oy i interface (with fhOi % 0.35). The additional term ð1 þ e11 Þð1 þ e22 Þ in Eq. (42) is introduced to correct for the area difference between the strained and unstrained crystalline oxide film, where e11 and e22 denote the (thickness-dependent) residual, homogeneous, normal strains in the oxide in the mutually perpendicular directions 1 and 2 along the hMijhMx Oy i interface plane at the growth temperature, T. The residual strains, eii , are related to the corresponding residual lattice spacings, i ahMx Oy i , of the hMxOyi film along the perpendicular directions 1 and 2 within the interface plane, by

ð42Þ


eii ¼

ahMx Oy i i ahMx Oy i ia hMx Oy i

ði ¼ 1; 2Þ

ð43Þ

strain , due to the state The strain energy contribution, chMijhM x Oy i of residual homogeneous strain in the crystalline oxide film at hhMx Oy i and T, is obtained from [41] ij eij ¼ h c strain ¼h r Cijkl eij ekl hMijhMx Oy i

hMx Oy i

hMx Oy i

ði; j; k; l ¼ 1; 2; 3Þ

ð44Þ

where rij is the stress tensor, Cijkl is the fourth-rank stiffness tensor, and eij is the residual homogeneous strain tensor of hMxOyi (with direction 3 perpendicular to the interface plane). dislocation Finally, the dislocation energy contribution, chMijhM , x Oy i at hhMOx i and T, is given by: 1 dislocation 2 dislocation dislocation þ ð45Þ chMijhM ¼ c c hMijhMx Oy i hMijhMx Oy i x Oy i dislocation dislocation and 2 chMijhM as the energies of the two with 1 chMijhM x Oy i x Oy i mutually perpendicular, regularly spaced arrays of misfit dislocations with Burgers vectors parallel to the two perpendicular directions 1 and 2 in hMx Oy i [41]. Different approaches for crystalline misfit accommodation at an interface (as reported in the literature) have been compared and evaluated in Ref. [41] to asses the misfit-endislocation . It was found that the so-called ergy contribution i chMijhM x Oy i “First Approximation” approach (APPR) of Frank and van der Merwe (for details, see Refs. [41, 106]) has the greatest dislocation overall accuracy for the estimation of i chMijhM x Oy i ðhhMx Oy i ; TÞ for a wide range of initial lattice-mismatch values in both the monolayer and nanometer thickness regimes (up to about ten oxide monolayers (ML); 1 ML*0.2 – 0.3 nm). If the oxide-film thickness exceeds 10 ML, the extrapolation approach (EXTR) of Frank and van der Merwe (for details, see Refs. [41, 106]) is also well dislocation ðhhMx Oy i ; TÞ. applicable for calculation of i chMijhM x Oy i Substitution of Eqs. (42), (44), and (45) in Eq. (36) then entropy & 0; see above): gives (assuming chMijhM x Oy i

chMijhMx Oy i ¼

1 fhOi D HO!hMi

OhOi

ð1 þ e11 Þ ð1 þ e22 Þ

þhhM O i Cijkl eij ekl x

y

dislocation 2 dislocation þ þ1 chMijhM c hMijhMx Oy i x Oy i

ð46Þ

dislocation Due to the thickness dependence of both eii and chMijhM x Oy i (and thus of chMijhMx Oy i ; see Eqs. (43 – 46)), the calculation

1295

F


of chMijhMx Oy i as a function of hhMx Oy i and T can only be performed numerically. To this end, the resultant interface energy, chMijhMx Oy i , according to Eq. (46) is minimized with respect to the residual homogeneous strain in the film for a given oxide-film thickness, hhMx Oy i , and growth temperature, T, by introducing 1 ahMx Oy i and 2 ahMx Oy i as the only fit parameters, values of which are obtained by requiring: qchMijhMx Oy i qeij

¼0

ð47Þ

Appropriate literature references providing elastic constants of metals and crystalline oxides, as required for the calculation of chMijhMx Oy i , have been listed in Refs. [3, 41]. The “unstrained” molar interface area, OhOi , (see Eq. (46)), as well as the “unstrained” lattice parameters, i ahMx Oy i , along the corresponding perpendicular directions 1 and 2 within the interface plane (see Eq. (43)), can be evaluated on the basis of known (i. e. experimentally observed) ORs between hMi and hMxOyi; one should generally refrain from adopting an average-crystal-plane value for OhOi , particularly for systems with a high metal–oxygen bond strength (see discussion below Eq. (36) in the introduction of Section 4.3.). Note that the differences between interaction interaction and chMijfM , as calculated using Eqs. (42) chMijhM x Oy i x Oy g and (32), arise only from differences in the values of OhOi and OfOg and the presence of residual strains, e11 and e22 , in the crystalline oxide overgrowth. Thus obtained values for the interfacial energy contributions strain dislocation due to strain (chMijhM ), misfit dislocations (chMijhM ) and x Oy i x Oy i interaction chemical interaction (chMijhMx Oy i ), as well as the value of the

(a)

(c)

1296

resultant hMijhMx Oy i interface energy (chMijhMx Oy i ), have been plotted in Fig. 9 as function of the oxide-film thickness (hhMx Oy i ) for the hNif111gijhNiOf100gi, the hZrf0001gijhZrO2 f111gi and the hTif1010gijhTiO2 f100gi interface. It follows that, because of the low initial lattice mismatch of only + 3 % and – 0.3 % along two mutually perpendicular directions parallel to the hTif1010gijhTiO2 f100gi interface plane (see Eq. (41)), all mismatch strain is accommodated fully elastically by the initial hTiO2 f100gi overgrowth on hTif1010gi (i. e. no misfit dislocations are built in at the metal/oxide interface at the onset of crystalline oxide growth). The associated strain energy contribution therefore increases linearly with increasing oxide-film thickness until a first array of misfit dislocations is introduced in the overgrowth at the metal/oxide interface along the highest mismatch direction for hhTiO2 i > 0.8 nm (compare Fig. 9a and b). The crystalline overgrowth of hNiOf100gi on hNif111gi, on the other hand, is associated with much higher initial lattice mismatches of + 19 % and + 3 % parallel to the hNif111gijhNiOf100gi interface plane. Consequently, the (anisotropic and tensile) elastic growth strain in the hNiOf100gi overgrowth becomes relaxed by the introduction of misfit dislocations at the hNif111gijhNiOf100gi interface already at the onset of growth along the high-mismatch direction and, subsequently, also along the lowmismatch direction (compare Fig. 9a and b). The release of tensile growth strain leads to a slight, favourable increase of the absolute value of the interaction energy contribution, interaction , due to the associated increase of the denchNif111gijhNiOf100gi sity of O–Ni bonds across the interface (Fig. 9c).

(b)

(d)

Fig. 9. (a) Strain energy contribution strain (chMijhM ), (b) misfit dislocation energy x Oy i dislocation contribution (chMijhM ), (c) interaction enx Oy i interaction ergy contribution (chMijhM ) and (d) resulx Oy i tant interfacial energy of the hMijhMx Oy i interface (chMijhMx Oy i ) as function of the oxidefilm thickness (hhMx Oy i ) for the hNif111gij hNiOf100gi, the hZrf0001gijhZrO2 f111gi and the hTif10 10gijhTiO2 f100gi interface [3] (For details, see Section 4.3.2.).


F

Feature


Analogously, if an initially compressive growth strain resides in the oxide overgrowth, as for hZrO2 f111gi on hZrf0001gi (with a corresponding near-isotropic initial lattice mismatch of – 5 %), the release of the elastic growth strain with increasing oxide thickness (by the introduction of misfit dislocations) is associated with an unfavourable decrease of the absolute value of the interaction energy contribution (Fig. 9c). It follows that, for compressively strained crystalline oxide overgrowths, generally a relatively larger part of the initial lattice mismatch can be accommodated elastically, as compared to tensilely strained crystalline oxide overgrowths (for metal–oxide systems with similar metal–oxygen bond strengths), because of the associated increase (by compression) of the density of metal-oxygen bonds across the interface, which makes the interaction energy more negative. This also implies that more elastic growth strain can be stored in the crystalline oxide overgrowth for metal/ oxide systems with a more negative interaction energy contribution, c interaction (i. e. a more negative value of D H 1 hMijhMx Oy i

O!hMi

in Eq. (46)). For most metal/oxide systems, the sum of the strain and dislocation energy contributions to the interface energy does not exceed the value of 0.5 J m – 2 (cf. Fig. 9a and b).

5. Ultrathin oxide overgrowths on metals Metal oxides, as functional materials, are applied in nanotechnologies, such as tunnel junctions [23 – 25], gas sensors [28, 29], model catalysts [30, 31], and (thin) diffusion barriers for corrosion resistance [26, 27]. The microstructure of these oxides often differ from those known and as predicted by bulk thermodynamics. For example, ultra-thin (< 3 nm) oxide films, nano-sized oxide particles or oxide nano-wires prepared by thermal or plasma oxidation of pure metals (or semiconductors) such as Al, Hf, Zr, Ta, Nb, Ge, and Si, are often amorphous, as long as the higher bulk energy of the amorphous oxide phase (as compared to that of the competing crystalline oxide phase) can be overcompensated by its lower sum of surface and interface energies (cf. Refs. [3, 108] and references therein; see also Section 2.2.). Only if the thickness of the amorphous oxide film exceeds a critical value, it can be transformed into a crystalline oxide film of same composition [2, 3, 56]. On the other hand, for the oxidation of metals as Cu, Co, Fe, Ni, Mo, and Zn, initial oxide overgrowth directly proceeds by nucleation and growth of a (semi)coherent, strained crystalline oxide film [3, 108]. In these cases, after attaining some critical oxidefilm thickness, the build-up growth strain in the oxide film is released by the formation of misfit dislocations (i. e., plastic deformation occurs), which dislocations are initiated at the metal/oxide interface [3, 41]: see Fig. 9 and related discussion in Section 4.3.2. Furthermore, crystalline oxide phases, metastable according to “bulk” thermodynamics, can be thermodynamically preferred by their relatively low surface energies: e. g., c-Al2O3 instead of a-Al2O3 [73], c-Y2O3 instead of a-Y2O3 [109] or tetragonal ZrO2 instead of monoclinic ZrO2 [110]. Also in these cases, only above some critical oxide-film thickness or particle size, transformation into a more stable (according to bulk thermodynamics) crystalline oxide phase can occur [2, 3]. Int. J. Mat. Res. (formerly Z. Metallkd.) 100 (2009) 10

Also, as predicted theoretically and found experimentally [42], an unexpected crystallographic orientation relationship (OR), characterized by a very high mismatch of an ultrathin crystalline oxide overgrowth and its parent metal surface, can occur. This OR is stabilized, with respect to the corresponding crystalline oxide overgrowth with a OR of the lowest mismatch, as a consequence of favourable interface energetics. The above mentioned experimental observations and supporting thermodynamic model predictions oppose many previous literature statements (e. g. see Refs. [111 – 113]) that the occurrence of an amorphous or pseudomorphic oxide phase on bare metal substrates, or the occurrence of unusual ORs, upon oxidation at low temperatures (of, say, T < 600 K) would be due to kinetic obstruction of the formation of the stable crystalline bulk modification. Obviously, fundamental and comprehensive knowledge on the thermodynamics of these nano-sized oxide microstructures is of utmost importance for the aforementioned nano-technologies: one strives for either a stable amorphous oxide phase or a stable coherent, single-crystalline (template) oxide phase, because of the absence of grain boundaries in both oxide-film modifications; such grain boundaries would act as paths for fast ionic or electronic migration, thereby deteriorating material properties such as the electrical resistivity, corrosion resistance or catalytic activity [28, 57, 114 – 116]. In particular for applications in the field of microelectronics, thin amorphous oxide films are required (e. g. a-SiO2, a-Al2O3, (a-HfO2)x(a-Al2O3)1–x, because of their uniform thickness and specific microstructure (no grain boundaries, moderate bond flexibility, large free volume, negligible growth strain) and related properties (e. g., passivating oxide-film growth kinetics, low leakage current, high dielectric constant, high corrosion resistance) [57, 115, 116]. In the following two typical cases of theoretical prediction and experimental verification of such nano-sized oxide microstructures are presented. The first example (Section 5.1.) deals with the thermodynamic stability of ultrathin amorphous oxide overgrowths on their metal substrates

Fig. 10. HRTEM micrograph and corresponding LEED pattern (at 100 eV) of the amorphous fAl2 O3 g overgrowth on hAlf111gi after oxidation for t = 6000 s at T = 373 K and pO2 = 1 · 10 – 4 Pa (and subsequent in-situ deposition of a MBE-grown Al seal after in-situ LEED analysis). The direction of the primary electron beam was along the ½11 2 zone axis of the hAlf111gi substrate (with the [111] direction perpendicular to the substrate-oxide interface) [56].

1297

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(a)

Fig. 11. Recorded oxide-film growth curves [15] for the thermal oxidation of bare hAlf111gi and hAlf100gi substrates at 350 K, 450 K, and 550 K (at pO2 = 1 · 10 – 4 Pa). The presented oxide-film growth curves were obtained by fitting experimental growth curves, as obtained by real-time in-situ spectroscopic ellipsometry (RISE), with theoretical growth curves as calculated on the basis of the coupled currents of cations and electrons (by both tunneling and thermionic emission) under a surface-charge field. See Ref. [15], for details. The grey area indicates the calculated critical thickness range, crit * 0.7 ± 0.1 nm, up to which an amorphous fAl2 O3 g overhfAl 2 O3 g growth (in instead of the corresponding crystalline hc-Al2 O3 i overgrowth) is predicted to be thermodynamically preferred on the parent hAlf111gi and hAlf100gi substrates [42, 56].

(b)

(as compared to the competing crystalline oxide). The second example (Section 5.2.) focuses on the origin of a OR of high lattice mismatch between a metal substrate and its ultra-thin (< 1 nm) oxide overgrowth. 5.1. Thermodynamically stable amorphous oxide films A HRTEM micrograph of an ultrathin amorphous fAl2 O3 g overgrowth on hAlf111gi after thermal oxidation for t = 6000 s at T = 373 K is shown in Figure 10. All Al2O3 films grown on hAlf111gi by thermal oxidation at T £ 450 K are amorphous, as determined by LEED and HRTEM, and have limiting, uniform thicknesses, hfAl2 O3 g , in the range of 0.7 to 0.8 nm; see Fig. 11 for hAlf111gi [15, 56]. The limiting thickness values of these evidently stable fAl2 O3 g overgrowths comply well with the predicted critical crit = 0.7 ± 0.1 nm (*3 – 4 oxide monolayers thickness of hfAl 2 O3 g with 1 ML*0.22 nm) up to which an amorphous fAl2 O3 g film is thermodynamically preferred on the hAlf111gi substrate (for T = 350 – 900 K; compare Figs. 11 and 12a). As follows from the thermodynamic model calculations according to the procedure outlined in Section 2.2., the fAl2 O3 g overgrowth on hAlf111gi is thermodynamically preferred with respect to the competing hc-Al2O3i overgrowth due to the slightly lower energy of the hAlð111ÞijfAl2 O3 g interface (because the corresponding hc-Al2O3i overgrowth is tensilely strained [3]; see Section 4.3.2.) in combination with a relatively small bulk energy difference between fAl2 O3 g and hc-Al2O3i. If a high activation energy barrier exists for the corresponding amorphous-to-crystalline transition, the initial fAl2 O3 g overgrowth may be maintained for oxide-film thickcrit . nesses beyond hfAl 2 O3 g Thermal oxidation of bare hAlf111gi substrates at more elevated temperatures T = 475 K instead results in the 1298

(c)

crit Fig. 12. Calculated critical thickness, hfM , up to which an amorx Oy g phous oxide overgrowth (instead of the corresponding crystalline oxide overgrowth) is thermodynamically preferred on the different low-index surfaces of (a) bare hAli metal substrates [2, 42, 56] (b) bare hZri metal substrates [3] and (c) bare hCri metal substrates [41], as function of the crit were determined by growth temperature (T). The values of hfM x Oy g solving hfMx Oy g according to Eq. (8) for DG cell ðhfMx Oy g ; TÞ = 0. For details, see Sections 2.2, 3.2., and 4.

formation of (epitaxial) crystalline hc(-like)-Al2O3i films crit = 0.7 ± 0.1 nm beyond the critical thickness, hfAl 2 O3 g (Figs. 11 and 12a). An HRTEM micrograph of a corresponding epitaxial hc(-like)-Al2O3i overgrowth on hAlf111gi for t = 6000 s at T = 475 K is shown in Fig. 13 (compare with Fig. 10). The predicted critical oxide thicknesses up to which various amorphous oxide overgrowths are stable with respect to their corresponding crystalline modifications on the most densely-packed surfaces of their metal substrates [2, 3, 41, 42, 104] have been plotted as a function of the growth temperature in Fig. 14. Exemplary dependencies of the calculated critical oxide-film thicknesses on the metal substrate orientation have been presented in Fig. 12 for oxide overgrowths on different low-index crystallographic surfaces of hAli, hCri, and hZri [3, 41]. The strikingly high stability of the fSiO2 g overgrowth on crit hSif111gi (i. e. hfSiO > 40 nm [3]) is due to the exception2g Int. J. Mat. Res. (formerly Z. Metallkd.) 100 (2009) 10

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Fig. 13. HRTEM micrograph and corresponding LEED pattern (at 53 eV) of the crystalline hc(-like)-Al2O3i overgrowth on hAlf111gi after oxidation for t = 6000 s at T = 373 K and pO2 = 1 · 10 – 4 Pa (and subsequent in-situ deposition of a MBE-grown Al seal after in-situ LEED analysis). The direction of the primary electron beam was along the ½11 2 zone axis of the hAlf111gi substrate (with the [111] direction perpendicular to the substrate-oxide interface). The inset shows the corresponding LEED pattern as recorded (with a primary electron energy of 53 eV) directly after the oxidation (prior to in-situ deposition of the Al seal); the six-fold symmetry observed in the LEED pattern is typical for the {111} surface of a crystalline oxide with an fcc-type oxygen sublattice, such as hc-Al2O3i [56].

the amorphous oxide-film configuration, the relatively large difference in bulk energy between the amorphous and crystalline oxide hinders a stabilization of the amorphous oxide phase beyond a thickness of 1 nm (*5 oxide MLs). For oxide overgrowths on hMgf0001gi and hNif111gi, the calculated critical oxide-film thickness is less than 1 oxide ML, which indicates that the development of a thermodynamically stable, amorphous oxide film on these metal surfaces is unlikely. Despite the relatively low energy of the hMgð0001ÞijfMgOg interface, the large difference in bulk energy between amorphous and crystalline MgO causes the critical thickness of the amorphous overgrowth on hMgð0001Þi to be that small. For the overgrowth on hNif111gi the (negative) surface and interface energy differences between amorphous and crystalline NiO are too small to compensate the corresponding (positive) bulk energy difference [3]. Oxide overgrowth on hCrf110gi, hCuf111gi; and hFef110gi is predicted to proceed by the direct formation and growth of a (semi)coherent crystalline oxide phase (i. e. crit < 0), in accordance with the limited number of exhfM x Oy g perimental observations reported in the literature [117 – 123]. In these cases the (negative) sum of the surface and interfacial energy differences of the amorphous and crystalline oxide overgrowths are too small to overcompensate the corresponding (positive) bulk energy difference. crit The dependence of hfM on the metal substrate orientax Oy g tion (Fig. 12) is mainly determined by differences in hMxOyi surface energy and M–O bond density across the hMijhMx Oy i interface for the differently oriented crystalline oxide overgrowths (as imposed by the OR between the crystalline oxide overgrowth and the parent metal substrate) [3]. 5.2. Thermodynamic stability of high-mismatch crystalline oxide films

crit Fig. 14. Calculated critical thickness (hfM ) up to which an amorx Oy g phous oxide overgrowth (instead of the corresponding crystalline oxide overgrowth) is thermodynamically preferred on the most densely packed face of the corresponding bare metal substrate as function of the growth temperature for various metal/oxide systems. The right ordinate indicates the corresponding critical thickness in oxide monolayers (MLs) as obtained by taking 1 oxide ML ffi 0.22 nm (see also Refs. [2, 3, 41, 42, 104]).

ally small bulk Gibbs energy difference between amorphous and crystalline SiO2 in combination with the considerably lower surface energy of amorphous fSiO2 g (as compared to hSiO2i; see also Fig. 6c) [3, 55]. Amorphous fSiO2 g films with thicknesses of several micrometers were found to exist up to temperatures as high as 1400 K [57], which hints at a high activation energy for the corresponding amorphous-to-crystalline transition. For oxide overgrowths on hZrf0001gi and hTif0001gi, in spite of the relatively low surface and interfacial energy for Int. J. Mat. Res. (formerly Z. Metallkd.) 100 (2009) 10

In-situ LEED and ex-situ HRTEM analysis [42] indicate the existence of a OR between the hc(-like)-Al2O3i overgrowth and the hAlf111gi substrate according to: hAlð111Þ½110ijjhc-Al2 O3 ð111Þ½110i, which is the expected OR with lowest possible mismatch (of about + 2.0 % at T = 300 K) between hAlf111gi and hc-Al2O3i. As for the thermal oxidation of bare hAlf111gi substrates, an overall stoichiometric Al2O3 film of uniform thickness develops on the bare hAlf100gi substrate after oxidation for 6000 s in the temperature range of 350 – 600 K [42]. For oxidation temperatures T < 450 K, the oxide films were found to be amorphous. For T ‡ 450 K, a crystalline hc(-like)-Al2O3i overgrowth develops on the hAlf100gi substrate (beyond an experimentally verified critical oxide-film thickness of about 0.45 ± 0.15 nm [56]) with a OR relationship according to [42]: hAlð100Þ½011ijjhc-Al2 O3 ð111Þ½011i (see Fig. 15). This OR corresponds to an initial lattice mismatch between hAlf100gi and hc-Al2O3i as large as + 18 % (at T = 300 K) in one direction parallel to the hAlð100Þijhc-Al2 O3 ð111Þi interface plane and a much lower initial lattice mismatch of about + 2.0 % (as for the overgrowth on hAlf111gi; see Sections 5.1. and Fig. 13) in the perpendicular direction. According to the LEED and HRTEM analysis, the large anisotropic tensile growth strain in the hc(-like)-Al2O3i overgrowth on hAlf100gi has predominantly been relaxed by the formation of defects at the incoherent hAlð100Þijhc-Al2 O3 ð111Þi interface, as well as by slight, in-plane rotations (of about ± 48) of 1299

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Fig. 15. HRTEM micrograph and corresponding LEED pattern (at 53 eV) of the high-mismatch crystalline hc(-like)-Al2O3i overgrowth on hAlf100gi after oxidation for t = 6000 s at T = 550 K and pO2 = 1 · 10 – 4 Pa. The direction of the primary electron beam was along the ½1 21 zone axis of the Al capping layer and the oxide film. The area of the micrograph within the square represents a Fourier-filtered region of the original micrograph as obtained after inverse Fourier transformation of the 2D Fourier transform of the original image after removing the noise around the primary beam spot. The inset shows the corresponding LEED pattern as recorded (with a primary electron energy of 53 eV) directly after the oxidation (prior to in-situ deposition of the Al seal), which shows the separate diffraction spots originating from the hAlf100gi substrate (exhibiting a four-fold symmetry) and the crystalline oxide overgrowth (exhibiting a twelve-fold symmetry with spots located in rings) [42].

Fig. 16. Calculated (a) residual strain energy, (b) misfit dislocation energy, and (c) chemical interaction energy contributions to (d) the resultant hAlijhc-Al2 O3 i interface energy, chAlijhAl2 O3 i , as function of the oxide film thickness, for hc-Al2O3i overgrowth on the bare hAlf100gi substrate at 298 K. The calculations were performed on the basis of the thermodynamic approach presented in Section 4.3.2., while adopting either the low-mismatch (i. e., hAlf100gijhc-Al2 O3 f100gi or the highmismatch (i. e., hAlf100gijhc-Al2 O3 f111gi OR between the oxide overgrowth and the hAlf100gi substrate. The corresponding energies for the overgrowth of low-mismatch (i. e., hAlf111gijhc-Al2 O3 f111gi on the hAlf111gi substrate are also shown for comparison [42].

1300

two types of hc(-like)-Al2O3i domains, which have their {111} plane parallel to the hAlf100gi surface, but are rotated with respect to each other by 908 around the surface normal [42]. The unexpected occurrence of a high lattice-mismatch OR between a hc(-like)-Al2O3i overgrowth and its parent hAlf100gi substrate can be explained considering the surface-energy and interface-energy contributions for the hc-Al2O3i overgrowth on hAlf100gi. Application of the thermodynamic procedures outlined in Sections 3.2.2. and 4.3.2., for both the observed case of high-mismatch OR between overgrowth and hAlf100gi, and for the originally expected case of low-mismatch OR between overgrowth and hAlf100gi, leads to the results shown in Fig. 16. It follows that, for both the low- and high-mismatch OR, the built-up elastic growth strain within the oxide overgrowth already gets released by the introduction of misfit dislocations within the monolayer thickness regime (Fig. 16a and b). The elastic-strain, misfit-dislocation, and interaction-energy contributions to the resultant interface energy (Fig. 16d), all attain approximately constant values at a thickness of about 1 nm. As expected, the energy contribution, mismatch chAlf100gijhc -Al2 O3 f111gi , due to the sum of the residual growth strain and misfit dislocations in the hc-Al2O3i overgrowth (Section 4.3.) is considerably larger (i. e. more positive) for the high-mismatch OR. However, the corresponding interinteraction action energy contribution, chAlf100gijhc -Al2 O3 f111gi , is much more negative (as compared interaction to chAlf100gijhc ) due to a higher density of metal-Al2 O3 f100gi oxygen bonds across the hAlð100Þijhc-Al2 O3 ð111Þi interface (than across the hAlð100Þijhc-Al2 O3 ð100Þi interface). interaction Since the (negative) interaction energy, chAlf100gijhc -Al2 O3 f111gi , is the dominant energy contribution to the interface energy (Section 4.3.), the larger (positive) mismatch contribution, mismatch mismatch chAlf100gijhc -Al2 O3 f111gi (as compared to chAlf100gijhc-Al2 O3 f100gi ), is overcompensated by its more negative interaction energy interaction contribution, chAlf100gijhc -Al2 O3 f111gi . In addition, the energy of the hc-Al2O3(111)i surface is much lower than that of the less-densely-packed hc-Al2O3(100)i surface (i. e. chcS;0-Al2 O3 ð111Þi % 0.9 J m – 2, whereas chcS;0-Al2 O3 ð100Þi ffi 1.9 J m – 2) [124]. Thus the observed high-mismatch OR for the initial hc-Al2O3i overgrowth on hAlf100gi is thermodynamically preferred (instead of the low-mismatch OR), because of the lower sum of the surface and interface energy contributions, in spite of the higher energy contributions due to residual strain and misfit dislocations in the corresponding high-mismatch hc-Al2O3i overgrowth. This implies that the generally adopted assumption, that the OR corresponding with the lowest possible lattice mismatch (i. e. the “best fit” OR) is energetically preferred, needs not hold for ultrathin overgrowths: the role of surface and interface energies can be dominant for the thermodynamic stability of the oxide film.

6. Metal-induced crystallization Amorphous semiconductors like silicon and germanium can crystallize at a temperature much lower than their “bulk” crystallization temperature when they are put in direct contact with a metal, such as Al [125], Au [126], Ag [127], Ni [128], Cu [129], and Pd [130] (Fig. 17). This phenomenon, which is now commonly referred to as metal-inInt. J. Mat. Res. (formerly Z. Metallkd.) 100 (2009) 10

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Fig. 17. The reduction in the crystallization temperature, T crys, of a-Si induced by contact with various metals [132].

duced crystallization (MIC), was firstly observed 40 years ago [131]. In the past decade, owing to its great potential for application in low-temperature fabrication of crystalline-Si-based (further designated as c-Si or hSii) thin-film devices such as flat-panel displays and solar cells on lowcost/flexible but usually heat-sensitive substrates, MIC has been extensively investigated in various metal/amorphoussemiconductor systems [17, 132]. The strong covalent bonding in bulk amorphous semiconductors accounts primarily for their high crystallization temperatures. At the interface with a metal layer, however, the covalent bonds become weakened, allowing for a relatively high mobility of the interfacial atoms, called “free” semiconductor atoms in the following. This layer of “free” semiconductor atoms is about 2 monolayers (ML) thick [133] and is generally believed to provide the agent for initiation of crystallization of amorphous semiconductors at low temperatures. By considering quantitatively the interface energetics related to these 2-ML interfacial “free” atoms (i. e. the competition between the change of the “bulk” energies and the change of the corresponding surface and interface energies) upon initiation of MIC, the different MIC temperatures/behaviours in various immiscible metal/amorphous-semiconductor systems can be understood and predicted on a unified basis. 6.1. Thermodynamics of grain-boundary wetting The metal layers employed to induce the crystallization of amorphous semiconductors are usually polycrystalline and possess a high grain-boundary (GB) density. These GBs in the metal layer might be wetted by the free semiconductor atoms in the contacted amorphous semiconductor layer and eventually mediate the MIC process. The possibility for the occurrence of this GB wetting process depends thermodynamically on whether the total interface energy can be reduced by replacing the GB with two interphase boundaries, no matter whether the wetting phase is liquid or solid [16, 134]. On the basis of the methods for calculation of various interface energies as function of T given in Sections 4.1., 4.2.1., and 4.3.1., the energetics for possible GB wetting processes in many crystalline-metal/amorphoussemiconductor (hMijfSg) systems have been evaluated quantitatively. As shown in Fig. 18, the energy of a highInt. J. Mat. Res. (formerly Z. Metallkd.) 100 (2009) 10

(a)

(b) Fig. 18. (a) Energetics for wetting of high-angle hAli grain boundaries (GBs) by a-Si and a-Ge. (b) Energetics for wetting of high-angle hAui GB 2 chMijfSg ) GBs and hAgi GBs by a-Si. Positive driving forces (chMi are evident for the occurrence of GB wetting in these systems [17].

angle GB in the crystalline metal, hMi, (i. e. the value of GB GB , cGB e. g. chAli hAgi or chAui , as evaluated on the basis of Eq. (39) in Section 4.3.1.) can indeed be substantially larger than the sum of interface energies of the two corresponding hMijfSg interfaces formed upon wetting of the high-angle GB in the metal layer by an amorphous semiconductor phase, fSg (i. e. larger than two times the value of e. g. chAlijfSig , chAlijfGeg , chAgijfSig or chAuijfSig , as evaluated using Eq. (30) in Section 4.2.1.) [17]. It follows that the wetting of the high-angle metal GBs by amorphous semiconductors can be favoured, because it reduces the total Gibbs energy of the system. This grain-boundary wetting process can play an important role in the initiation of MIC (see what follows). 6.2. Thermodynamics of nucleation of crystallization Metal-mediated nucleation of crystalline semiconductors at low temperatures could occur heterogeneously at the interface with the metal and/or at the wetted metal GBs (Section 6.1.). A factor obstructing nucleation of crystallization at these interfaces and at wetted GBs is that the energy of the created crystalline/crystalline interface(s) is usually higher than that of the original crystalline/amorphous inter1301

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face(s) (see Sections 4.2. and 4.3.) [1, 2]. Consequently, analogously to the oxidation of metals (Sections 2.2. and 5.1.), a thin amorphous semiconductor film at the interface(s) or within the GB of a contacting crystalline metal layer can be thermodynamically stable up to a certain critical thickness; beyond this critical thickness, the higher “bulk” Gibbs energy of the amorphous phase is no longer overcompensated by its lower sum of interface energies. Furthermore, it is important to recognize, as a further constraint, that the critical thickness should be smaller than 2 MLs in order that crystallization can initiate at a crystalline-metal/amorphous-semiconductor interface at relatively low temperatures, because the layer of “free” semiconductor atoms at the interface has a maximal thickness of only 2 ML (see the introductory part of Section 6). crit , for the nucleation of a The critical thickness, hhMijfSg crystalline semiconductor, hSi, at the hMijfSg interface with a crystalline metal, hMi, can be evaluated as a function of the temperature, T, by dividing the increase of interface energy accompanying crystallization (in J m – 2) by the corresponding decrease of the “bulk” Gibbs energy (as given crystallization by the bulk crystallization energy, DGhSi!fSg , in J m – 3): crit hhMijfSg ðTÞ ¼

chMijhSi ðTÞ þ chSijfSg ðTÞ chMijfSg ðTÞ crystallization DGhSi!fSg ðTÞ

ð48Þ

crit It is important to recognize that hhMijfSg ðTÞ should be smaller than (or maximally equal to about) 2 ML (see above). Alternatively, crystallization of the semiconductor could also initiate at initially-wetted, high-angle GBs in the contacting metal (see above). The critical thickness for the crystallization of the corresponding wetting film of “free” fSg at the metal GBs is given by: h i 2 chMijhSi ðTÞ chMijfSg ðTÞ crit ð49Þ ðTÞ ¼ hhMijfSgjhMi crystallization DGhSi!fSg ðTÞ

The wetting fSg film at the metal GBs is sandwiched between two hMi grains and, consequently, the maximum thickness of “free” fSg that can wet the metal GBs at low temperatures is *2 · 2 ML = 4 ML (Section 6.1.). Hence, crit ðTÞ must be smaller than (or maximally equal hhMijfSgjhMi to) 4 ML in order that crystallization of the wetting fSg film can initiate at the metal GBs at the concerned temperature. The critical thicknesses for initiation of crystallization at the original hMijfSg interfaces, as well as at the wetted hMijfSgjhMi GBs in hMi, have been calculated as a function of T for various metal/amorphous-semiconductor systems using Eqs. (48) and (49), respectively: see Fig. 19. The required values for the crystalline–amorphous and crystalline–crystalline interface energies (i. e. values of chMijfSg and chMijhSi ) were evaluated as a function of T using Eqs. (30) and (40) in Sections 4.2.1. and 4.3.1., respectively. For example, the calculated critical thicknesses for initiation of crystallization of a-Si at the hAlijfSig interface with hAli, as well as at its wetted GBs, are plotted as a function of temperature in Fig. 19a. It follows that the calculated critical thickness for crystallization of a-Si at the hAlijfSig interface is larger than 2 ML up to 400 8C and beyond. This implies that initiation of Al-induced crystallization at the hAlijfSig interface 1302

is thermodynamically impossible. At the hAli GBs for T > 140 8C, on the other hand, the critical thickness for crystallization of the wetting a-Si film is below 4 ML. Hence, for the hAli–fSig layer system, the only site for c-Si to nucleate at low temperatures is the Al GB with a predicted temperature for the onset of crystallization > 140 8C. Indeed, experimental studies of the MIC process in hAli–fSig layer systems have indicated a minimal temperature for the onset of crystallization of 150 8C [17, 135]. It has also been confirmed experimentally that the MIC process in hAli–fSig layer systems is initiated exclusively at the Al GBs and not at the original hAlijfSig interface [10, 125]. Similar theoretical results for the Al/a-Ge bilayer system are shown in Fig. 19a as well. It follows that the initiation of crystallization of a-Ge in hAli–fGeg layer systems can occur both at the hAlijfGeg interface and at the Al GBs (for

(a)

(b) Fig. 19. (a) Calculated critical thicknesses for nucleation of c-Si and cGe at the hAli GBs and at the and interfaces. Note that the thicknesses of the “free” (or fGeg) layers are about 2 ML at the interfaces with hAli and *4 ML at the hAli GBs. Both these thicknesses are shown by grey, horizontal lines in the figure. It follows that c-Si can only nucleate at the hAli GBs at (above) *140 8C, and that c-Ge can nucleate both at the GBs and at the hAlijfGeg interface above *50 8C. (b) Calculated critical thicknesses for nucleation of c-Si at hAgi and hAui GBs and at hAgijfSig and hAuijfSig interfaces. It follows that c-Si can nucleate at the hAgi GBs at (above) *400 8C. For the hAui–fSig system, it follows that c-Si cannot nucleate directly at hAui GBs and at the hAuijfSig interface. Instead, MIC in hAui–fSig layer systems is mediated by the formation of metastable hAu3 Sii phase nucleated at hAui GBs [17].


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T > 50 8C). This prediction is fully consistent with corresponding experimental observations [10, 17]. Corresponding theoretical results for the initiation of crystallization in hAgi–fSig and hAui–fSig bilayer systems are compared in Fig. 19b. For the hAgi–fSig system, the initiation of crystallization of a-Si is predicted to proceed exclusively at the hAgi GBs and only for T > 400 8C. This predicted MIC behavior also agrees very well with experimental observations of MIC in the Ag/a-Si layer system [127, 135]. For the hAui–fSig system, the calculation shows that c-Si cannot nucleate directly both at the hAuijfSig interface and at the GBs (Fig. 19b). Instead, experiments have shown that the MIC process in the hAui–fSig system is mediated by the formation of a metastable hAu3 Sii silicide phase at a very low temperature of *100 8C [126]. Therefore, thermodynamic calculations, which account for the nucleation of hAu3 Sii at the hAuijfSig interface and at the hAui GBs, have also been carried out (Fig. 19b). Indeed, the theoretical critical thickness for the formation of hAu3 Sii at fSig-wetted hAui GBs is equal or lower than 4 ML for T > 80 8C, which is well compatible with the observed formation of hAu3 Sii at hAui GBs in the hAui–fSig system at *100 8C [126].

(a)

6.3. Continued crystallization After the formation of a hSi nucleus at a hMi GB, the wetted GB in the metal layer is replaced by two hMijhSi interphase boundaries. To continue the crystallization process of fSg, the atoms in the original fSg layer now need to diffuse into the hMijhSi interphase boundaries (“wetting”) and crystallize there. The driving force for this secondary wetting process is given by: DcfSg!hMijhSi ¼ chMijhSi ðchMijfSg þ chSijfSg Þ

ð50Þ

This driving force is calculated to be positive for continued crystallization in the hAli–fSig system (Fig. 20a), where initial nucleation of c-Si occurs exclusively at the hAli GBs (Section 6.2.), which implies that “free” fSig atoms (see introduction of Section 6) are capable to continue to wet the hAlijhSii boundaries. Once wetting fSg films have been formed at the hMijhSi interphase boundaries, the following two processes can be considered: (i) the “wetting” fSg layer joins with the adjacent hSi grains to crystallize, as a result of which the hSi grains grow laterally, i. e. perpendicular to the hMijhSi boundaries, and/or (ii) new grains of hSi nucleate at the wetted hMijhSi boundaries. Ad (i): the critical thickness for continued, lateral grain growth of hSi perpendicular to the hMijhSi boundaries is given by: h i chMijhSi ðTÞ chMijfSg ðTÞ þ chSijfSg ðTÞ crit ð51aÞ hhSi grain growth ¼ crystallization DGhSi!fSg ðTÞ

crit hhSi new nucleation ¼

(b) Fig. 20. (a) Energetics of the continued diffusion of “free” fSig atoms into the sublayer after completing the initial nucleation of c-Si at the hAli GBs. A positive driving force is predicted for the continued wetting of the hAlijhSii boundaries by fSig. (b) Energetics for continued lateral grain growth of c-Si in the original hAli layer (perpendicular to the original hAli GBs). Continued grain growth is favored, whereas the formation of new c-Si nuclei is impossible [17].

Ad (ii): the critical thickness for the formation of new hSi grains at the hMijhSi boundaries, is given by Eq. (51b): (see bottom of page) The calculated critical thicknesses for the hAli–fSig system according to Eqs. (51a) and (51b) are plotted as a function of temperature T in Fig. 20b. It follows that the critical thickness for the formation of new c-Si nuclei according to Eq. (51b) is as large as *4 ML. Recognizing that the thickness of the “free” fSig atoms adjacent to hAli metal is only about 2 ML, it follows that the formation of new c-Si nuclei at the hAlijhSii boundaries is impossible at low temperatures. Instead, the critical thickness for continued c-Si grain growth according to Eq. (51a) is only *1.5 ML (at T > 150 8C). Hence, continued lateral growth of the c-Si grains initially formed at the original hAli GBs is possible,

h i chMijhSi ðTÞ þ chSijhSi ðTÞ chMijfSg ðTÞ þ chSijfSg ðTÞ crystallization ðTÞ DGhSi!fSg


ð51bÞ

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which is in accordance with the in-situ and ex-situ TEM analyses. Since the driving force, DcfSg!hMijhSi , for Si atoms diffusing into the hAlijhSii boundaries is also positive (see Fig. 20a) continued lateral growth of the initially nucleated c-Si grains at the original hAli GBs is possible, indeed. The continuous inward diffusion and crystallization (lateral grain growth) of Si within the original Al GBs eventually results in a layer exchange of the Al and Si sublayers (for discussion, see Ref. [17] and references therein). For those metal/amorphous-semiconductor systems in which the nucleation of crystallization can initiate both at the interface with metal and at the metal GBs (e. g. in hAli–fGeg layer systems), the MIC process does not involve a layer exchange [10, 136].

(a)

(b) Fig. 21. (a) Schematic representation of the energetics for the initiation of crystallization of a-Si at hAli GBs in an ultrathin, columnar hAli overlayer. (b) Calculated critical thickness for initiation of crystallization of a-Si at hAli GBs as functions of both the hAli overlayer thickness, hhAli , critical and the temperature, T. The line pertaining to hhAlijfSigjhAli = 4 ML gives a theoretical prediction of the crystallization temperature of a-Si as function of the hAli overlayer thickness. The corresponding experimental confirmation of the dependence of the crystallization temperature of aSi on hhAli has also been indicated [4, 132].

6.4. Ultrathin metal-induced crystallization Consider the nucleation of hSi at metal GBs, which have been initially wetted by fSg (Sections 6.1. and 6.2.). If the thickness of the original metal sublayer, hMi, is that small that it is comparable to the thickness of the fSg wetting film, the energetics for nucleation of hSi at the hMi GBs is not correctly described by Eq. (49): see the schematic illustration for the hAlijfSig layer system in Fig. 21a. For such small values of the Al overlayer thickness, hhAli , and the wetting fSig film, hfSig , it follows that upon initiation of crystallization of the wetting fSig film at the hAli GBs (taken as running perpendicular to the film surface: columnar grain structure), not only the interface energy change, 2 hhAli ðchAlijhSii chAlijfSig Þ, associated with the replacement of the two original hAlijfSig interfaces by two hAlijhSii interfaces, but also the surface and interface enS S cfSig Þ and hfSig chSiijfSig , asergy changes, hfSig ðchSii sociated with the formation of the crystalline hSii surface and the hSiijfSig interface, respectively, have to be considered (see Fig. 21a). Consequently, the critical thickness for initiation of crystallization of a-Si at the Al GBs as function of both T and hhAli is given by [4] (Eq. (52)): (see bottom of page) crit The critical thickness, hhAlijfSigjhAli , as function of both hhAli and T, as calculated according to Eq. (52), is presented in the contour plot of Fig. 21b. For the hAlijfSig system, the only possible sites for initiation of crystallization of a-Si are the hAli GBs (Section 6.2.) and therefore the calculated crit must be smaller than (or maximally be value of hhAlijfSigjhAli equal to) about 4 ML in order that MIC can occur. Thus the thermodynamic prediction of the temperature for the onset of crystallization of the wetting fSig film at the hAli GB, as function of hAli overlayer thickness, is given by crit = 4 ML in Fig. 21b. the solid line pertaining to hhAlijfSigjhAli It follows that the crystallization temperature is around 150 – 200 8C for hAli overlayer thicknesses hhAli > 20 nm. For hhAli < 20 nm, the crystallization temperature increases strongly with decreasing hhAli . These theoretical predictions have been experimentally confirmed by monitoring the crystallization behavior of a-Si as a function of the hAli overlayer thickness under ultra-high vacuum conditions by real-time in-situ spectroscopic ellipsometry [4]. The corresponding crystallization temperature of a-Si decreases from about 700 8C to 180 8C for an increase of the thickness hhAli of the covering Al film from hhAli < 1 nm to hhAli = 20 nm (Fig. 21b).

7. Conclusion The contributions of surface and interface energies to the total Gibbs energy can predominate the energetics of lowdimensional systems. On the basis of this recognition the formation of many experimentally observed, nano-sizerelated microstructures, which are in flagrant contrast

h i 2 chAlijhSii ðTÞ chAlijfSig ðTÞ crit h i hhAlijfSigjhAli ðT; hAl Þ ¼ S S c ðTÞ c ðTÞ þ chSiijfSig ðTÞ hSii fSig crystallization ðTÞ DGhSii!fSig hhAli 1304

ð52Þ


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with expectations derived from bulk phase diagrams, can be understood on a purely thermodynamic basis. Powerful expressions, straightforwardly applicable to practical cases, for the estimation of Gibbs energies of solid surfaces and solid–solid heterointerfaces between crystalline and amorphous metals, semiconductors, and oxides have been obtained on the basis of the macroscopic atom approach. The above thermodynamic modeling leads to predictions as (i) the formation of stable, amorphous solid–solution phases at metal–metal interfaces, (ii) the formation of stable, amorphous oxide phases at the surface of metal substrates and (iii) “wetting” of grain boundaries by an amorphous phase. The proposed thermodynamic model description also provides quantitative estimates for the thickness of these amorphous product layers beyond which crystallization of a stable crystalline phase should occur. All these predictions are in (quantitative) agreement with experimental observations. The power of the presented thermodynamic analysis of interface (and surface) energies is in particular illustrated by the prediction, and experimental verification, of the strong temperature dependence of the metal-induced crystallization of a semiconductor as Si by thickness variation of the adjacent crystalline metal layer. We are indebted to Prof. Dr. F. Sommer for helpful discussion on the estimation of interface energies between metals and semiconductors. We are grateful to our former co-worker and colleague Dr. J.Y. Wang for his cooperation in the original studies on metal-induced crystallization. We thank Dr. G. Richter for HRTEM analysis of the ultra-thin oxide overgrowths and Dr. P. A. van Aken for provision of TEM facilities. References [1] R. Benedictus, A. Bottger, E.J. Mittemeijer: Phys. Rev. B 54 (1996) 9109. DOI:10.1103/PhysRevB.54.9109 [2] L.P.H. Jeurgens, W.G. Sloof, F.D. Tichelaar, E.J. Mittemeijer: Phys. Rev. B 62 (2000) 4707. DOI:10.1103/PhysRevB.62.4707 [3] F. Reichel, L.P.H. Jeurgens, E.J. Mittemeijer: Acta Mater. 56 (2008) 659. DOI:10.1016/j.actamat.2007.10.023 [4] Z.M. Wang, J.Y. Wang, L.P.H. Jeurgens, E.J. Mittemeijer: Phys. Rev. Lett. 100 (2008) 125503. DOI:10.1103/PhysRevLett.100.125503 [5] E. Arzt: Acta Mater. 46 (1998) 5611. DOI:10.1016/S1359-6454(98)00231-6 [6] X. Batlle, A. Labarta: J. Phys. D. Appl. Phys. 35 (2002) R 15. [7] A.D. Yoffe: Adv. Phys. 51 (2002) 799. DOI:10.1080/00018730110117451 [8] U. Dahmen, S. Hagège, F. Faudot, T. Radetic, E. Johnson: Philos. Mag. 84 (2003) 2651. DOI:10.1080/14786430410001671403 [9] Q.S. Mei, K. Lu: Prog. Mater. Sci. 52 (2007) 1175. DOI:10.1016/j.pmatsci.2007.01.001 [10] Z.M. Wang, J.Y. Wang, L.P.H. Jeurgens, E.J. Mittemeijer: Scripta Mater. 55 (2006) 987. DOI:10.1016/j.scriptamat.2006.08.029 [11] P. Lejcek, S. Hofmann: Crit. Rev. Solid State Mat. Sci. 33 (2008) 133. DOI:10.1080/10408430801907649 [12] F. Reichel, L.P.H. Jeurgens, E.J. Mittemeijer: Phys. Rev. B 73 (2006) 024103. DOI:10.1103/PhysRevB.73.024103 [13] H. Over: Prog. Surf. Sci. 58 (1998) 249. DOI:10.1016/S0079-6816(98)00029-X [14] G.A. Somorjai, M.A. Vanhove: Prog. Surf. Sci. 30 (1989) 201. DOI:10.1016/0079-6816(89)90009-9 [15] F. Reichel, L.P.H. Jeurgens, E.J. Mittemeijer: Acta Mater. 56 (2008) 2897. DOI:10.1016/j.actamat.2008.02.031 [16] G.A. Lopez, E.J. Mittemeijer, B.B. Straumal: Acta Mater. 52 (2004) 4537. DOI:10.1016/j.actamat.2004.06.011


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(Received July 14, 2008; accepted July 29, 2009)


Bibliography DOI 10.3139/146.110204 Int. J. Mat. Res. (formerly Z. Metallkd.) 100 (2009) 10; page 1281 – 1307 # Carl Hanser Verlag GmbH & Co. KG ISSN 1862-5282

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