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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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.
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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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L. P. H. Jeurgens et al.: Thermodynamics of reactions and phase transformations at interfaces and surfaces
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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L. P. H. Jeurgens et al.: Thermodynamics of reactions and phase transformations at interfaces and surfaces
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,
Int. J. Mat. Res. (formerly Z. Metallkd.) 100 (2009) 10
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L. P. H. Jeurgens et al.: Thermodynamics of reactions and phase transformations at interfaces and surfaces
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].
Int. J. Mat. Res. (formerly Z. Metallkd.) 100 (2009) 10
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
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L. P. H. Jeurgens et al.: Thermodynamics of reactions and phase transformations at interfaces and surfaces
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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L. P. H. Jeurgens et al.: Thermodynamics of reactions and phase transformations at interfaces and surfaces
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].
Int. J. Mat. Res. (formerly Z. Metallkd.) 100 (2009) 10
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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Feature
L. P. H. Jeurgens et al.: Thermodynamics of reactions and phase transformations at interfaces and surfaces
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
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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
Int. J. Mat. Res. (formerly Z. Metallkd.) 100 (2009) 10
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Feature
L. P. H. Jeurgens et al.: Thermodynamics of reactions and phase transformations at interfaces and surfaces
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
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L. P. H. Jeurgens et al.: Thermodynamics of reactions and phase transformations at interfaces and surfaces
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
Int. J. Mat. Res. (formerly Z. Metallkd.) 100 (2009) 10
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L. P. H. Jeurgens et al.: Thermodynamics of reactions and phase transformations at interfaces and surfaces
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
Int. J. Mat. Res. (formerly Z. Metallkd.) 100 (2009) 10
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L. P. H. Jeurgens et al.: Thermodynamics of reactions and phase transformations at interfaces and surfaces
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
Int. J. Mat. Res. (formerly Z. Metallkd.) 100 (2009) 10
interaction
chCrijfCr
,
2 O3 g
ð34Þ
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Feature
L. P. H. Jeurgens et al.: Thermodynamics of reactions and phase transformations at interfaces and surfaces
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
<AB> 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
Int. J. Mat. Res. (formerly Z. Metallkd.) 100 (2009) 10
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L. P. H. Jeurgens et al.: Thermodynamics of reactions and phase transformations at interfaces and surfaces
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
Int. J. Mat. Res. (formerly Z. Metallkd.) 100 (2009) 10
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
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L. P. H. Jeurgens et al.: Thermodynamics of reactions and phase transformations at interfaces and surfaces
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.).
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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].
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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
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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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L. P. H. Jeurgens et al.: Thermodynamics of reactions and phase transformations at interfaces and surfaces
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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L. P. H. Jeurgens et al.: Thermodynamics of reactions and phase transformations at interfaces and surfaces
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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L. P. H. Jeurgens et al.: Thermodynamics of reactions and phase transformations at interfaces and surfaces
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].
Int. J. Mat. Res. (formerly Z. Metallkd.) 100 (2009) 10
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L. P. H. Jeurgens et al.: Thermodynamics of reactions and phase transformations at interfaces and surfaces
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
Int. J. Mat. Res. (formerly Z. Metallkd.) 100 (2009) 10
ð51bÞ
1303
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L. P. H. Jeurgens et al.: Thermodynamics of reactions and phase transformations at interfaces and surfaces
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
Int. J. Mat. Res. (formerly Z. Metallkd.) 100 (2009) 10
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L. P. H. Jeurgens et al.: Thermodynamics of reactions and phase transformations at interfaces and surfaces
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.
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(Received July 14, 2008; accepted July 29, 2009)
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Bibliography
DOI 10.3139/146.110204
Int. J. Mat. Res. (formerly Z. Metallkd.)
100 (2009) 10; page 1281 – 1307
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