Thermodynamics of reactions and phase transformations at interfaces and surfaces

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27 Οκτ 2013 (πριν από 3 χρόνια και 9 μήνες)

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Lars P.H.Jeurgens,
a
Zumin Wang,
a
Eric J.Mittemeijer
a,b
a
Max Planck Institute for Metals Research,Stuttgart,Germany
b
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 reac-
tions and phase transformations at interfaces between met-
als,semiconductors,oxides and the ambient have been re-
viewed.Unanticipated nanostructures,characterized by the
presence of phases at interfaces and surfaces which are un-
stable 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 realis-
tic,practically and generally applicable thermodynamic
model descriptions of microstructural evolutions at inter-
faces and surfaces have been outlined.To this end,expres-
sions for the estimation of the involved interface and sur-
face energies have been dealt with extensively as a
function of,e.g.,the filmcomposition and the growth tem-
perature.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-conduc-
tors in contact with various metals.
Keywords:Interface energy;Surface energy;Thermody-
namics;Transformations;Nanomaterials;Thin films;Amor-
phous solids.
1.Introduction
The thermodynamics of reactions and phase transformations
in nanomaterials,with their characteristically high interface
density,can deviate significantly from “expected” behav-
iours for “bulk” materials,e.g.as derived from“bulk” phase
diagrams [1–4].Thin film systems (with (sub)layer thick-
nesses in the nanometer range) provide classical examples
of such nanomaterials.Obviously,the relatively large vol-
ume fractions of atoms associated with interfaces and sur-
faces in such low-dimensional systems (i.e.with one dimen-
sion,the layer thickness,expressed in the size scale of atoms)
can bring about energy contributions,which activate mech-
anisms 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 nano-
sized system not only has pronounced consequences for the
energetics of the system:dimensional and microstructural
constraints occur by disturbing the lattice periodicity,there-
by confining the mobilities of e.g.photons,phonons,plas-
mons and/or dislocations [5–7].In general,low-dimensional
systems,such as thin films and sheet assemblies (2-dimen-
sional 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 corre-
sponding 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 electri-
cal,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 filmsystems.
Typical thermodynamic driving forces for microstruc-
tural transitions in thin filmsystems in contact with the am-
L.P.H.Jeurgens et al.:Thermodynamics of reactions and phase transformations at interfaces and surfaces
Int.J.Mat.Res.(formerly Z.Metallkd.) 100 (2009) 10 1281
bient (i.e.vacuum,a gas atmosphere or an adsorbed layer)
are the lowering of surface,interface,and/or grain-bound-
ary energies by (impurity) segregation effects [11,12],ad-
sorbate-induced surface reconstructions [13–15],wetting
[4,16–18] and/or interfacial mixing or compound forma-
tion [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 ki-
netics for thermodynamic equilibration,which makes arti-
ficially,man-made thin filmsystems prone to degradation.
Controlling the thermodynamic stability of solid–solid in-
terfaces between metals,alloys,semiconductors,oxides,
biomaterials,and the ambient is therefore of cardinal im-
portance 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],met-
al/oxide contacts in microelectronics and photovoltaic de-
vices [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 fre-
quently applied,evasive invocation of a “kinetic” constraint
to “understand” the experimental observation of unantici-
pated (nano)structures,which differ from those known and
predicted by bulk thermodynamics.
For example,experimental observations of the forma-
tion of amorphous alloy phases by interdiffusion at inter-
faces and grain boundaries of crystalline multilayers in
e.g.the Ni–Ti,Cu–Ta,Al–Pt,and Mg–Ni system (a pro-
cess commonly referred to as solid-state amorphisation;
SSA) have previously been rationalized on the basis of cri-
teria 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.Conse-
quently,thin amorphous films developing at the interface
(surface) and/or grain boundaries can in principle be ther-
modynamically 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–cera-
mic heterointerfaces and metal–oxide interfaces (with typi-
cal equilibriumthicknesses 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.
To satisfy the technological demand for control of the
thermodynamic stability and related properties of low-di-
mensional 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 trans-
formations at interfaces and surfaces should be modeled as
function of the material and operating conditions,such as
the film thickness,the chemical composition and constitu-
tion,the temperature and operation time,as well as the am-
bient conditions.Up to date,such generally applicable de-
scriptions 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 atomapproach [1–3,40–43],ori-
ginally proposed and developed by Miedema and co-work-
ers [44–46] (see Sections 2 and 4 for details).
The present paper provides a detailed overview of recent
accomplishments in the aforementioned modeling of reac-
tions 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 de-
sign of a thermodynamic model,specifying the essential
energy contributions,is discussed and illustrated in Sec-
tion 2.The needed expressions for the assessment of solid
surface Gibbs energies of amorphous and crystalline met-
als,semiconductors,and oxides as a function of the tem-
perature are provided in Section 3.Expressions for the esti-
mation of the Gibbs energies of heterointerfaces between
crystalline and amorphous metals,semiconductors and ox-
ides,as a function of the temperature and the size of the sys-
tem (as given by the film thickness) are presented in Sec-
tion 4.Finally,comparison of thermodynamic predictions,
on the above basis,with experimental observations is pre-
sented 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 belowtheir bulk crystallization temperature (a pro-
cess commonly referred to as metal-induced crystalliza-
tion).
2.Basis of thermodynamic analysis;identification of
energy contributions
Thermodynamic analysis of a phase transformation begins
with the identification and evaluation of the involved driv-
ing force(s).The total Gibbs energy change of a systemac-
companying a phase transformation from state A to state B
is given by:DG
tot
A!B
¼ G
tot
B
G
tot
A
(Fig.1).The thermody-
namic driving force is defined as the negative of the total
Gibbs energy change:DG
tot
A!B
,i.e.a positive driving
force exists if the phase transformation is associated with
a lowering of the system’s total Gibbs energy (i.e.
DG
tot
A!B
¼ G
tot
B
G
tot
A
< 0).A thermodynamically desired
phase transformation (DG
tot
A!B
¼ G
tot
B
G
tot
A
< 0) can be
hindered by kinetic barriers expressed by a single rate-lim-
iting activation energy,as Q
A!B
in Fig.1,or by several (a
series of) rate-determining [47,48]) activation energies,
which may be overcome by thermal activation.For exam-
L.P.H.Jeurgens et al.:Thermodynamics of reactions and phase transformations at interfaces and surfaces
1282 Int.J.Mat.Res.(formerly Z.Metallkd.) 100 (2009) 10
F
Feature
ple,in the case of an amorphous-to-crystalline solid-state
phase transformation [3,18],the thermal energy of the in-
volved 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 crystal-
line phase.Or,analogously,the solid-state wetting of grain
boundaries (as driven by a lowering of the total grain-
boundary 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 gov-
erned 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 coun-
teracted in thin film systems by energy penalties for the
creation of crystalline surfaces and crystalline/crystalline
interfaces from the original amorphous surfaces and crys-
talline/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 filmsystems,involves spe-
cification of all energy contributions to a phase transforma-
tion.This non-trivial set-up of an appropriate thermody-
namic 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 in-
terdiffusion 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 thermody-
namic analysis of such a phase transformation will be de-
scribed for a unit cell of volume ½ h
hAi
þh
hBi
 per unit inter-
face area,as defined in Fig.2a,where h
hAi
and h
hBi
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
G
cell
hAijhBi
¼ h
hAi

G
hAi
V
hAi
þh
hBi

G
hBi
V
hBi
þ2  c
hAijhBi
ð1Þ
where G
hAi
and G
hBi
are the Gibbs energies (per mole) and
V
hAi
and V
hBi
the molar volumes of hAi and hBi,respec-
tively,and c
hAijhBi
denotes the interface energy of the
hAijhBi interface (per unit area).After formation of thin pro-
L.P.H.Jeurgens et al.:Thermodynamics of reactions and phase transformations at interfaces and surfaces
Int.J.Mat.Res.(formerly Z.Metallkd.) 100 (2009) 10 1283
F
Feature
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 trans-
formation:i.e.DG
tot
B!A
¼  G
tot
B
G
tot
A
 
.The transformation can
be kinetically hindered at lowtemperatures by the associated activation
energy barrier,Q
A!B
.
1
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 thermody-
namics of formation of crystalline and amor-
phous 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
h
hAi
and h
hBi
of the initial crystalline hAi and
hBi metal layers,respectively.See Sec-
tion 2.1.for details.
duct layers of the crystalline hABi solid solution at the origi-
nal hAijhBi interfaces (with uniform thicknesses,h
hABi
),the
total Gibbs energy of the defined unit cell is given by
(Fig.2b)
G
cell
hAijhABijhBi
¼ ðh
hAi
h
hABi!hAi
Þ 
G
hAi
V
hAi
þðh
hBi
h
hABi!hBi
Þ

G
hBi
V
hBi
þðh
hABi!hAi
þh
hABi!hBi
Þ 
G
hABi
V
hABi
þ2
 ðc
hAijhABi
þc
hBijhABi
Þ ð2aÞ
where
1
2
 h
hABi!hAi
and
1
2
 h
hABi!hBi
are the thicknesses of
the hABi layers grown in layer hAi and hBi,respectively (with
h
hABi
¼
1
2
 h
hABi!hAi
þ
1
2
 h
hABi!hBi
;see Fig.2b;G
hABi
and
V
hABi
represent the Gibbs energy and molar volume of the
hABi solid solution;and c
hAijhABi
and c
hBijhABi
denote the in-
terface 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 config-
uration is given by:
DG
f
hABi
 G
hABi
x
A
 G
hAi
ð1 x
A
Þ  G
hBi
ð2bÞ
where x
A
denotes the mole fraction of A in hABi.Pro-
vided that the molar volumes of the components hAi and
hBi do not significantly change upon alloying (i.e.
V
hABi
¼ x
A
 V
hAi
þð1 x
A
Þ  V
hBi
),Eq.(2a) can be re-
written as
G
cell
hAijhABijhBi
¼ h
hAi

G
hAi
V
hAi
þh
hBi

G
hBi
V
hBi
þ2  h
hABi

DG
f
hABi
V
hABi
þ2  ðc
hAijhABi
þc
hBijhABi
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
DG
CSS
¼ G
cell
hAijhABijhBi
G
cell
hAijhBi
¼ 2  h
hABi

DG
f
hABi
V
hABi
þ2  ðc
hAijhABi
þc
hBijhABi
c
hAijhBi
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 in-
terfaces (a process commonly referred to as solid-state
amorphisation;SSA) a similar expression for the associated
Gibbs energy change,DG
SSA
,results:
DG
SSA
¼ G
cell
hAijfABgjhBi
G
cell
hAijhBi
¼ 2  h
fABg

DG
f
fABg
V
fABg
þ2  ðc
hAijfABg
þc
hBijfABg
c
hAijhBi
Þ ð3bÞ
where h
fABg
,DG
f
fABg
,and V
fABg
denote the total thickness,
the Gibbs energy of formation and the molar volume of the
amorphous {AB} product layer,respectively;c
hAijfABg
and
c
hBijfABg
are the energies of the interfaces per unit area be-
tween 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 so-
lution is taken the same as that of the corresponding crystal-
line hABi solid solution of identical composition (i.e.
V
fABg
ffi V
hABi
).
The driving forces for the formation of crystalline (i.e.
DG
CSS
according to Eq.(3a)) and amorphous (DG
SSA
ac-
cording to Eq.(3b)) solid solution product layers at the origi-
nal 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,x
A
),
provided that corresponding expressions for the bulk Gibbs
energies of formation (i.e.DG
f
hABi
and DG
f
fABg
) and the ener-
gies of the interfaces between the various crystalline and
amorphous phases (c
hAijhBi
,c
hAijhABi
,c
hBijhABi
,c
hAijfABg
,and
c
hBijfABg
) are assessable as a function of T and x
A
.Experimen-
tal values and procedures for the assessment of DG
f
hABi
and
DG
f
fABg
as a function of T and x
A
are provided in,e.g.,Refs.
[1,43,44,46,49–51].Generally applicable formulations
for the calculation of the crystalline–amorphous and crys-
talline–crystalline interface energies as a function of T and
x
A
are presented in this paper (Sections 4.2.1.and 4.3.1.,re-
spectively).
An example of such thermodynamic model predictions
for the occurrences of crystalline and amorphous solid so-
lution 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 nm
2
and with individual layer thick-
nesses of 10 nm;see Fig.2) is provided by Fig.3.The cal-
culated interface energies,c
hNiijhNiTii
,c
hTiijhNiTii
,c
hNiijfNiTig
,
and c
hTiijfNiTig
(per unit interface area (symbol c),as well
as per volume of the defined unit cell (symbol C);see Sec-
tions 4.2.1.and 4.3.1.) have been plotted as a function of
x
Ni
in Fig.3a.The bulk Gibbs energies of formation,
DG
f
hNiTii
and DG
f
fNiTig
are shown (also as a function of
x
Ni
) in Fig.3b.It follows that the crystalline–amorphous
interface energies (i.e.c
hNiijfNiTig
and c
hTiijfNiTig
) are al-
ways lower than the corresponding crystalline–crystalline
energies (c
hNiijhNiTii
and c
hTiijhNiTii
) (Fig.3a).Consequently,
the amorphous {NiTi} product layer can be thermodynami-
cally 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.
less negative:DG
f
fNiTig
> DG
f
hNiTii
;see Fig.3b) is overcom-
pensated by its relatively lower sum of crystalline–amor-
phous interface energies (i.e.½c
hNiijfNiTig
þc
hTiijfNiTig
 <
½c
hNiijhNiTii
þc
hTiijhNiTii
).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].
Atheoretical value for the critical thickness,h
crit
fNiTig
,up to
which the amorphous fNiTig product layer is thermo-
dynamically,rather than kinetically (cf.Section 1),pre-
ferred (as compared to the competing crystalline hNiTii
product layer of identical composition) is obtained by solving
DG
CSS
hNiTii
ðh
hNiTii
;x
Ni
;TÞ ¼ DG
SSA
fNiTig
ðh
fNiTig
;x
Ni
;TÞ for h
fNiTig
for a given mole fraction,x
Ni
,of Ni in NiTi (employing
Eqs.(3a) and (3b) with V
fNiTig
ffi V
hNiTii
;see above):
h
crit
fNiTig
ðx
Ni
;TÞ
¼
ðc
hNiijhNiTii
þc
hTiijhNiTii
Þ ðc
hNiijfNiTig
þc
hTiijfNiTig
Þ
ðDG
f
fNiTig
DG
f
hNiTii
Þ=V
fNiTig
ð4Þ
For h
fNiTig
> h
crit
fNiTig
,bulk energy contributions become
dominant and the product layer will strive for crystalliza-
tion as a crystalline solid solution (or as a crystalline inter-
metallic compound).The calculated critical thickness,
L.P.H.Jeurgens et al.:Thermodynamics of reactions and phase transformations at interfaces and surfaces
1284 Int.J.Mat.Res.(formerly Z.Metallkd.) 100 (2009) 10
F
Feature
h
crit
fNiTig
,has been plotted as a function of x
Ni
in Fig.4.It fol-
lows that h
crit
fNiTig
has a maximum value of about 23 nm
for x
Ni
= 0.56 (at 525 K).The theoretical value of
h
crit
fNiTig
*10 nm for x
Ni
= 0.50 agrees very well with the ex-
perimentally determined thickness of 9.0 ± 0.2 nm of the
amorphous Ni
50
Ti
50
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].
2.2.Oxide formation at metal surfaces
Consider the formation of either an amorphous or a crystal-
line oxide overgrowth on a bare (i.e.without a native
oxide) single-crystalline metal substrate,hMi.The ener-
getics of an amorphous oxide overgrowth,{M
x
O
y
},with
uniform thickness,h
fM
x
O
y
g
,can be compared with those of
the competing crystalline oxide overgrowth,hM
x
O
y
i,of
equivalent uniformthickness,h
hM
x
O
y
i
[2].Again the braces,
{ },and angle brackets,h i,refer to the amorphous state and
the crystalline state,respectively.The competing fM
x
O
y
g
and hM
x
O
y
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 hMijfM
x
O
y
g
and hMijhM
x
O
y
i configurations will be compared for unit
cells of volumes h
fM
x
O
y
g
 l
2
fM
x
O
y
g
and h
hM
x
O
y
i
 l
2
hM
x
O
y
i
,re-
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 sche-
matic drawings in Fig.5.
The accumulation of elastic growth strain in the amor-
phous fM
x
O
y
g overgrowth (and the metal substrate) can
be neglected due to the relative large free volume and mod-
erate 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 hM
x
O
y
i overgrowth,on
the other hand,the initial lattice mismatch between
hM
x
O
y
i and hMi,which is governed by the crystallographic
orientation relationship (OR) of the (semi-)coherent
hM
x
O
y
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 fM
x
O
y
g and strained hM
x
O
y
i unit
cells contain the same molar quantity of oxygen reactant,
it holds that
l
2
fM
x
O
y
g
 h
fM
x
O
y
g
V
fM
x
O
y
g
¼
l
2
hM
x
O
y
i
 h
hM
x
O
y
i
X
hM
x
O
y
i
 V
hM
x
O
y
i
ð5aÞ
where V
fM
x
O
y
g
and V
hM
x
O
y
i
are the molar volumes of strain-
free fM
x
O
y
g and strain-free hM
x
O
y
i,respectively;l
fM
x
O
y
g
and l
hM
x
O
y
i
correspond to the widths and lengths in perpendi-
cular directions along the interface plane of the unstrained
fM
x
O
y
g unit cell and the strained hM
x
O
y
i unit cell,respec-
tively (see Fig.5).The fraction X
hM
x
O
y
i
on the right-hand side
of Eq.(5) relates the volume,V
str
hM
x
O
y
i
,as occupied by one
mole M
x
O
y
in the strained hM
x
O
y
i overgrowth,to the molar
volume of strain-free hM
x
O
y
i:i.e.V
str
hM
x
O
y
i
¼ X
hM
x
O
y
i

V
hM
x
O
y
i
.Furthermore,the corresponding ratios of the
heights (i.e.thicknesses) and surface areas of the un-
strained fM
x
O
y
g cell and strained hM
x
O
y
i cell are given
L.P.H.Jeurgens et al.:Thermodynamics of reactions and phase transformations at interfaces and surfaces
Int.J.Mat.Res.(formerly Z.Metallkd.) 100 (2009) 10 1285
F
Feature
(a)
(b)
(c)
Fig.3.Thermodynamic predictions for the formation of amorphous
and crystalline solid solution product layers at the original hNiijhTii in-
terfaces of a Ni–Ti multilayer by interdiffusion at 525 K,as calculated
for individual layer thicknesses of h
hNii
= h
hTii
= 10 nm and a unit cell
of lateral area 10 · 10 nm
2
and (see Fig.2).(a) The calculated crystal-
line–crystalline and crystalline–amorphous interface energies as func-
tion of the Ni fraction,x
Ni
,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 x
Ni.
(c) The calculated free energy change for solid-state amorphiza-
tion at the interfaces of the Ni–Ti multilayer as a function x
Ni
(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].
Fig.4.Calculated critical thickness,h
crit
fNiTig
,up to which an amorphous
fNiTig solid solution product layer is thermodynamically preferred
with respect to the corresponding crystalline hNiTii solid-solution pro-
duct layer as a function of the molar fraction of Ni for interface amor-
phisation in Ni–Ti multilayers at 525 K [1].
by [2,41]
n ¼
h
hM
x
O
y
i
h
fM
x
O
y
g
¼ ð1 þ
￿
e
33
Þ 
V
hM
x
O
y
i
V
fM
x
O
y
g
!
1
3
ð5bÞ
and
v ¼
l
hM
x
O
y
i
l
fM
x
O
y
g
!
2
¼ ð1 þ
￿
e
11
Þ  ð1 þ
￿
e
22
Þ 
V
hM
x
O
y
i
V
fM
x
O
y
g
!
2
3
ð5cÞ
respectively,where
￿
e
ij
represents the residual,homoge-
neous strain tensor of the hM
x
O
y
i overgrowth with the cor-
responding perpendicular directions 1 and 2 parallel to the
hMijhM
x
O
y
i interface plane and the direction 3 perpendicu-
lar 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)
G
cell
fM
x
O
y
g
¼ l
2
fM
x
O
y
g


h
fM
x
O
y
g

G
fM
x
O
y
g
V
fM
x
O
y
g
þc
S
fM
x
O
y
g
þc
hMijfM
x
O
y
g
!
ð6aÞ
and (Fig.5b)
G
cell
hM
x
O
y
i
¼ l
2
hM
x
O
y
i


h
hM
x
O
y
i

G
hM
x
O
y
i
X
hM
x
O
y
i
 V
hM
x
O
y
i
þc
S
hM
x
O
y
i
þc
hMijhM
x
O
y
i
!
ð6bÞ
where G
fM
x
O
y
g
and G
hM
x
O
y
i
are the bulk Gibbs energies per
mole fM
x
O
y
g and hM
x
O
y
i;c
S
fM
x
O
y
g
and c
S
hM
x
O
y
i
represent
the surface energies (per unit area) of the fM
x
O
y
g and
hM
x
O
y
i overgrowths in contact with the ambient (e.g.va-
cuum,a gas atmosphere or an adsorbed layer);and
c
hMijfM
x
O
y
g
and c
hMijhM
x
O
y
i
are the energies (per unit area) of
the interfaces between the metal substrate and the fM
x
O
y
g
and hM
x
O
y
i oxide overgrowth,respectively.The Gibbs en-
ergy of formation,DG
f
M
x
O
y
,of one mole M
x
O
y
oxide phase
out of its elements in their stable configuration,at a given
temperature and pressure,is defined as
DG
f
fM
x
O
y
g
 G
fM
x
O
y
g
x  G
hMi

y
2
 G
O
2
ðgÞ
ð7Þ
By employing Eqs.(5a),(5c),(6a),(6b),and (7),it then fol-
lows that the thermodynamic stability of the amorphous
oxide overgrowth with respect to that of the competing
crystalline oxide overgrowth,as expressed by the differ-
ence in total Gibbs energy of the corresponding fM
x
O
y
g
and hM
x
O
y
i unit cells (Fig.5),is given by
DG
cell
¼ G
cell
fM
x
O
y
g
G
cell
hM
x
O
y
i
¼ h
fM
x
O
y
g

DG
f
fM
x
O
y
g
DG
f
hM
x
O
y
i
V
fM
x
O
y
g
!
þc
S
fM
x
O
y
g
þc
hMijfM
x
O
y
g
v  ðc
S
hM
x
O
y
i
þc
hMijhM
x
O
y
i
Þ ð8Þ
Thus if DG
cell
ðh
fM
x
O
y
g
;TÞ < 0 the amorphous oxide over-
growth is more stable,whereas for DG
cell
ðh
fM
x
O
y
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
DG
f
hM
x
O
y
i
of a crystalline oxide phase will always be lower
than DG
f
fM
x
O
y
g
of the corresponding amorphous oxide phase
[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
be stable up to a certain critical thickness,h
crit
fM
x
O
y
g
[2,3,41,56].
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 over-
growth on the same metal),is obtained by solving h
fM
x
O
y
g
in
Eq.(8) for DG
cell
ðh
fM
x
O
y
g
;TÞ = 0 [2,3,41].To this end,ver-
satilely applicable formulations for the bulk,surface,and in-
terface energy terms of the fM
x
O
y
g and hM
x
O
y
i overgrowths,
as a function of the oxide-filmthickness,growth temperature,
metal substrate orientation,and its OR with the (semi-)coher-
ent hM
x
O
y
i overgrowth are required.These are presented in
Sections 3.2.,4.2.2.,and 4.3.2.If the resulting value of
h
critical
fM
x
O
y
g
is negative,it is implied that the oxide overgrowth
L.P.H.Jeurgens et al.:Thermodynamics of reactions and phase transformations at interfaces and surfaces
1286 Int.J.Mat.Res.(formerly Z.Metallkd.) 100 (2009) 10
F
Feature
(a)
(b)
Fig.5.Schematic drawing of competing amorphous and crystalline
oxide overgrowths of uniformthicknesses (<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 homoge-
neous amorphous oxide overgrowth,fM
x
O
y
g,of uniform thickness,
h
fM
x
O
y
g
,on the metal substrate.(b) the competing crystalline oxide
overgrowth,hM
x
O
y
i,of uniform thickness,h
h M
x
O
y
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.
Furthermore,the defined unit cells of volume h
fM
x
O
y
g
 l
2
fM
x
O
y
g
and
h
hM
x
O
y
i
 l
2
hM
x
O
y
i
,as indicated in (a) and (b) contain the same molar
quantity of fM
x
O
y
g and hM
x
O
y
i,respectively.
on the bare metal substrate is thermodynamically predicted to
proceed by the direct formation and growth of a (semi-)coher-
ent crystalline oxide phase.
The calculated bulk,interface,and surface energy differ-
ences,as well as the corresponding total Gibbs energy differ-
ence,DG
cell
¼ G
cell
fM
x
O
y
g
 G
cell
hM
x
O
y
i
,(see corresponding terms
in Eq.(8)) for competing amorphous and crystalline MgO,
TiO
2
and SiO
2
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 £ h
fM
x
O
y
g
£ 3 nm.The correspond-
ing 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-
sa in Fig.6a and b,respectively;i.e.h
crit
fMgOg
%0.2 nm and
h
crit
fTiO
2
g
*0.8 nm.Amorphous {SiO
2
} is the preferred initial
oxide overgrowth far beyond the largest film thickness con-
sidered in the present calculations (i.e.h
crit
fSiO
2
g
> 40 nm [3]),
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 indi-
cates that the development of a,thermodynamically stable,
amorphous oxide filmon bare Mg{0001} metal surfaces is un-
likely.The results of these critical thickness calculations are
compared in further detail with experimental data in Sec-
tion 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,c
S
A
,(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 sur-
face area,i.e.
c
S
A
ðTÞ ¼
H
S
A
ðTÞ T  S
S
A
ðTÞ
O
A
ðTÞ
ð9aÞ
where H
S
A
and S
S
A
are the excess enthalpy and the excess en-
tropy of the defined system due to the presence of the sur-
face O
A
in the defined system.
2
The temperature depen-
dence of the surface energy,qc
S
A
=qT (e.g.in J m
– 2
K
–1
),
is mainly governed by the surface entropy and thermal ex-
pansion (or shrink) [44,58,59].Consequently,neglecting
the individual temperature dependencies of H
S
A
,S
S
A
and
O
A
,the resulting temperature dependence of the surface en-
ergy,qc
S
A
=qT,is given by
qc
S
A
=qT ffi S
S
A
=O
A
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 contri-
butions is measured [60–65].For a single-component sys-
tem,the (excess) surface stress tensor,g
S
ij
,(which corre-
sponds to the reversible work required to produce unit area
of new surface by elastical stretching) is related to its sur-
face energy,c
S
(i.e.the excess Gibbs energy per unit sur-
face area,as defined by Eq.(9a)),according to [60,64–67]:
g
S
ij
¼ d
ij
 c
S
þ
qc
S
qe
S
ij
!
ð10Þ
where d
ij
is the Kronecker delta and e
S
ij
the elastic surface (ex-
cess) strain tensor.As follows fromEq.(10),the difficulty in
L.P.H.Jeurgens et al.:Thermodynamics of reactions and phase transformations at interfaces and surfaces
Int.J.Mat.Res.(formerly Z.Metallkd.) 100 (2009) 10 1287
F
Feature
(a)
(b)
(c)
Fig.6.Calculated bulk,interfacial and surface energy differences,
as well as the corresponding total Gibbs energy difference
(DG
cell
¼ G
cell
fM
x
O
y
g
G
cell
hM
x
O
y
i
;see Eq.(8) in Section 2.2.),of the com-
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 (h
fM
x
O
y
g
) at a growth temperature of
T
0
= 298 K [3].
2
Eq.(9a) considers a solid phase A with a homogeneous (bulk) com-
position up to its surface with the ambient and,consequently,any com-
positional variations at its outer surface by e.g.surface segregation
effects are not accounted for.Such compositional effects can be
accounted for by adding an additional term,
P
C
j¼1
l
j
 C
j
,to Eq.(9a),
where C denotes the total number of components in the system
and the term C
j
corresponds to the surficial excess of component j
per unit interface area (with a corresponding chemical potential,l
j
)
(cf.Refs.[11,67]).
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 contri-
butions to the surface energy:i.e.qc
S
=qe
S
ij
¼ 0 and thus
g
S
ij
¼ d
ij
 c
S
.In this case (i.e.for qc
S
=qe
S
ij
¼ 0),the notion
surface energy,c
S
,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 experimen-
tal determination of liquid and solid surface energies [60–
64,66].
Most experimentally determined values of,in particular
solid,surface energies are affected by significant experi-
mental uncertainties and errors.It is much easier to experi-
mentally assess accurate values for the surface energy,c
S;m
A
,
of the liquid at the melting point,T
m
,as derived by e.g.ses-
sile 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,c
S
A
ðT < T
m
Þ,of amorphous and polycrystalline
(elemental and homogenous compound) surfaces by extrapo-
lation from the surface energy of the corresponding liquid
phase at its melting point (provided that an experimental va-
lue for c
S;m
A
is available),according to
c
S
A
ðTÞ ¼ c
S;m
A
þqc
S
A
=qT  ðT T
m
Þ ð11aÞ
Experimental values for surface-orientation-specific ener-
gies of crystalline solid surfaces have only very scarcely
been reported in the literature (especially for single-crystal-
line compound phases).Therefore,if such orientation-spe-
cific surface energies are required (e.g.for single-crystal-
line metal or oxide microstructures such as oxides;see
Section 3.2.2.),but unavailable from the literature,esti-
mated values may be obtained departing from theoretical
values for such surface orientation-specific energies at
0 K,c
S;0
A
,as derived for “relaxed” (i.e.reconstructed) crys-
tallographic surfaces at 0 K (by first-principle or molecular
dynamics calculations).Next,extrapolation to the tempera-
ture of investigation can be performed according to (analo-
gously to Eq.(11a))
c
S
A
ðTÞ ¼ c
S;0
A
þqc
S
A
=qT  T ð11bÞ
The needed value for qc
S
A
=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 avail-
able experimental data or estimated on the basis of the
macroscopic atomapproach.Details are provided in the fol-
lowing Sections 3.1.1.and 3.2.1.
3.1.Surface energies of (semi-)metals
3.1.1.Amorphous (semi-)metal surfaces
The surface energy,c
S
fAg
,(per unit area) of an undercooled
(i.e.configurationally frozen) (semi-)metallic liquid phase,
{A},(as a model for the solid amorphous phase of the metal
or semiconductor) at temperature T can be expressed by
(see Eqs.(11a) and (9b))
c
S
fAg
ðTÞ ¼ c
S;m
fAg
þqc
S
fAg
=qT  ðT T
m
Þ
ffi c
S;m
fAg
S
S
fAg
=O
fAg
 ðT T
m
Þ ð12Þ
where c
S;m
fAg
is the surface energy of the corresponding liquid
phase at its melting point,T
m
.
Comprehensive experimental data sets for c
S;m
fAg
have
been reported in,e.g.,Refs.[58,59,63,68,70].The molar
surface area,O
fAg
,in Eq.(12) (S
S
fAg
now is defined as the
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
{A} surface atoms.Since the values of c
S;m
fAg
,S
S
fAg
,and
O
fAg
in Eq.(12) are intrinsically isotropic,the molar sur-
face area O
fAg
can be directly related (on the basis of the
macroscopic atom approach [44,46]) to the molar volume,
V
fAg
,of {A} at temperature T by:
O
fAg
¼ f
fAg
 C
0
 V
fAg
 
2
3
ð13Þ
where f
fAg
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 con-
stant C
0
relates the surface area of one mole atomic (i.e.
Wigner–Seitz) cells of the solid to its bulk volume (i.e.the
termC
0
 V
2=3
fAg
equals the sumof 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.f
fAg
¼
1
6
) and a sphere
(i.e.f
fAg
¼
1
2
),it follows that [44,46] the fraction f
fAg

1
3
and the proportionality constant C
0
= 4.5∙ 10
8
mol
–1/3
.
The surface entropy for amorphous metals (and semicon-
ductors) roughly equals S
S
fAg
*7.34 J mol
– 1
K
–1
[43]
(note:the value of S
S
fAg
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
estimates for qc
S
fAg
=qT ffi S
S
fAg
O
fAg

can be obtained
using S
S
fAg
*7.34 J mol
– 1
K
–1
and by adopting a value of
O
fAg
as calculated according to Eq.(13).Alternatively,
some system-specific,experimental values for qc
S
fAg
=qT
can be taken from Refs.[58,59,63,68].The negative tem-
perature dependence of the surface energy (i.e.qc
S
fAg
=qT ffi
S
S
fAg
=O
fAg
< 0) is approximately the same for the liquid
and the corresponding solid amorphous phase [44,58,59].A
rough empirical estimate for qc
S
fAg
=qT ð  qc
S
hAi
=qTÞ is:
–1:5
0:6
∙ 10
– 4
J m
2
K
–1
[44,58,59] (see also Section 4.2.2.).
Thus by adopting estimated or experimental values of
qc
S
fAg
=qT (see above),straightforward application of Eq.(12)
is possible to determine the surface energy,c
S
fAg
ðTÞ,of solid
amorphous (semi-)metals at any given T from experimental
values of the surface energy,c
S;m
fAg
,of phase {A} at its melting
point.
3.1.2.Crystalline (semi-)metal surfaces
The (orientation-specific) surface energy,c
S
hAi
,(per unit area)
of a crystalline solid,hAi,at temperature T,can be expressed
by (see Eqs.(11b) and (9b))
c
S
hAi
ðTÞ ffi c
S;0
hAi
þqc
S
hAi
=qT T ffi c
S;0
hAi
T  S
S
hAi
=O
hAi
ð14Þ
L.P.H.Jeurgens et al.:Thermodynamics of reactions and phase transformations at interfaces and surfaces
1288 Int.J.Mat.Res.(formerly Z.Metallkd.) 100 (2009) 10
F
Feature
where the quantities in this equation pertaining to the solid
crystalline phase hAi are defined analogously to those for
the solid amorphous phase {A} in Eq.(12).Values of c
S;0
hAi
and O
hAi
in Eq.(14) are generally anisotropic for crystalline
solid surfaces (i.e.dependent on the crystallographic orien-
tation of the considered surface plane),whereas the surface
entropy,S
S
hAi
,is approximately independent of the crystal
surface orientation (and the temperature;cf.Eq.(11b))
[44,59].
As discussed with respect to Eq.(11b) above,if reliable
crystallographic-specific values for c
S
hAi
ðTÞ are not avail-
able from experiment,theoretical estimates for the orienta-
tion-specific surface energies at 0 K,c
S;0
hAi
,may be applied.
Such theoretical values of c
S;0
hAi
for the low-index crystallo-
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 ex-
perimental values for qc
S
hAi
=qT as follows.
Some experimental,“average-crystal-plane” values of
qc
S
hAi
=qT for certain crystalline metals (and semi-conductors)
have been reported in e.g.Refs.[59,63,68]
(qc
S
hAi
=qTð qc
S
fAg
=qTÞ  1:5 ± 0:6∙ 10
– 4
J m
2
K
– 1
).Alter-
natively,a “crystal-plane-specific” estimate of qc
S
hAi
=qT ffi
S
S
hAi
=O
hAi
can be obtained by using S
S
hAi
*7.72 J mol
– 1
K
– 1
[43] in Eq.(14) and by adopting an orientation-specific val-
ue of O
hAi
,as approximated by the projected area enclosed
by one mole of hAi atoms allocated to a corresponding crys-
tallographic plane of hAi parallel to the surface [2,3,41].
It follows that values of c
S
hAi
ðTÞ are lower for more den-
sely packed crystallographic surfaces (e.g.the non-recon-
structed {111},{110},and {0001} surface planes for fcc,
bcc,and hcp metals,respectively),as well as that
c
S
hAi
> c
S
fAg
[44,58,59,69,71].
Alternatively,for polycrystalline solid surfaces,an
“average-crystal-plane” value for c
S;0
hAi
ðTÞ can be obtained
by extrapolation from the corresponding experimental va-
lue at the melting point,c
S;m
fAg
,after its multiplication with
an empirical correction factor of 1.13 (recognizing the
higher density of the crystalline phase,i.e.V
hAi
< V
fAg
and that f
hAi
> f
fAg
[59]).A corresponding “average-crys-
tal-plane” estimate for qc
S
hAi
=qT ffi S
S
hAi
O
hAi

can be ob-
tained using S
S
hAi
*7.72 J mol
– 1
K
–1
(see above) in combi-
nation with an “average-crystal-plane” value for O
hAi
taken
as (cf.Eq.(13)):
O
hAi
¼ f
hAi
 C
0
 ½V
hAi

2
3
ð15Þ
with f
hAi
= 0.35 [44,59] and C
0
= 4.5∙ 10
8
mol
–1/3
.
3.2.Surface energies of oxides
3.2.1.Amorphous oxide surfaces
Estimated surface energies,c
S
fM
x
O
y
g
,of solid amorphous ox-
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 en-
ergy,c
S;m
fM
x
O
y
g
,of the liquid oxide at its melting point,T
m,
using an experimental or estimated value for the tempera-
ture dependence,qc
S
fM
x
O
y
g
=qT (see below).
Some experimental values of c
S;m
fM
x
O
y
g
have been reported
in,e.g.,Refs.[70,72,73].Alternatively,an estimate of
c
S;m
fM
x
O
y
g
can be obtained from an established empirical rela-
tionship between c
S;m
fM
x
O
y
g
and the molar volume,V
m
fM
x
O
y
g
,
of the oxide phase at T
m
(Fig.7) [74]:
c
S;m
fM
x
O
y
g
ffi1:764 k
B
T
m

V
m
fM
x
O
y
g
N
A
 x
 

2
3
0:0372 ðJ  m
2
Þ ð16Þ
where x is the number of metal ions per M
x
O
y
unit “mole-
cule” (k
B
and N
A
denote Boltzman’s constant and Avoga-
dro’s constant,respectively).
The surface energies,c
S
fM
x
O
y
g
,of molten oxides typically
have only a very weak negative temperature dependence
with an averaged value of about qc
S;m
fM
x
O
y
g
=qT*–0.7 ± 0.5∙
10
– 4
J m
2
K
–1
[3] (which is lower than the corresponding
empirical estimate for metals and semiconductors of
qc
S
hAi
=qT  qc
S
fAg
=qT*–1.5 ± 0.6∙ 10
–4
J m
2
K
– 1
;see
Sections 3.1.1.and 3.1.2.).Otherwise,a rough estimate for
qc
S
fM
x
O
y
g
=qT ffi S
S
fM
x
O
y
g
=O
fM
x
O
y
g
ðTÞ can be obtained using
S
S
fM
x
O
y
g
*7.34 J mol
– 1
K
–1
(see Section 3.2.1.).
Finally,it is noted that,only for some (molten) network-
forming oxides (e.g.GeO
2
,B
2
O
3
,and V
2
O
5
),surprisingly,
a very weak positive temperature dependence of c
S
fM
x
O
y
g
has
been found:qc
S
fM
x
O
y
g
=qT*+0.4 ± 0.3∙ 10
– 4
J m
– 2
K
– 1
(see
references listed in Ref.[74]).
3.2.2.Crystalline oxide surfaces
Experimental values for the orientation-specific surface en-
ergies of crystalline oxides are extremely scarce (only litera-
ture 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 es-
timates of c
S
hM
x
O
y
i
ðTÞ are generally determined by extrapola-
tion (see Eq.(11b)) from corresponding theoretical values
for c
S;0
hM
x
O
y
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-
L.P.H.Jeurgens et al.:Thermodynamics of reactions and phase transformations at interfaces and surfaces
Int.J.Mat.Res.(formerly Z.Metallkd.) 100 (2009) 10 1289
F
Feature
Fig.7.Surface energy,c
S;m
fM
x
O
y
g
,of liquid oxides at their melting point,T
m
,
versus the corresponding energy term,k
B
 T
m
 ½V
m
fM
x
O
y
g
=ðN
A
 xÞ
2=3
.
The dashed line represents a linear fit through the data points according
to Eq.(16) in Section 3.2.1.[3,74].
ues in Ref.[3]).Alternatively,a rough estimate of c
S;0
hM
x
O
y
i
for
any low-index crystalline oxide surface can be obtained from
an established empirical relationship [3]:
c
S;0
hM
x
O
y
i
ffi 0:0105  N
NN
MO
 E
lattice
hM
x
O
y
i
 y
1
ð17Þ
where N
NN
MO
denotes the molar number of broken,nearest-
neighbour bonds at the oxide surface per unit surface area
(which depends on the crystallographic surface plane con-
sidered),E
lattice
hM
x
O
y
i
is the lattice energy (i.e.the Gibbs energy
to form the oxide from its respective ions at T = 0 K) and y
is the number of oxygen ions per M
x
O
y
molecule.An (even)
more accurate empirical relationship for c
S;0
fM
x
O
y
g
has been
established for the {100},{110} and {111} crystallo-
graphic faces of the oxide phases with a rock salt structure
(Fig.8) [3],i.e.
c
S;0
hMOfhklgi
ffi ￿ E
lattice
hMOi
 V
0
hMOi

2
3
 N
A

1
3
ð18Þ
with U = 0.012,U = 0.026,and U = 0.028 for the {100},
{110},and {111} crystallographic faces,respectively (V
0
hMOi
denotes the molar volume of the oxide at T = 0 K).Further-
more,the surface energy (at T = 0 K) of high-index oxide sur-
faces 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,
qc
S;m
hM
x
O
y
i
=qT,can be taken equal to that for the corre-
sponding amorphous oxide phase (Section 3.2.2.),or else a
rough “orientation-specific” estimate can be obtained
from qc
S
hM
x
O
y
i
=qT ffi S
S
hM
x
O
y
i
=O
hM
x
O
y
i
with S
S
hM
x
O
y
i
*
7.72 J mol
–1
K
–1
(see Section 3.2.1.).
4.Assessment of solid–solid interface energies
In analogy with the definition of the solid surface energy
(Section 3.1.),the energy,c
AjB
,per unit area of the interface
between two homogeneous solid phases A and B at tem-
perature T can be defined as the excess Gibbs energy of
the atoms or molecules of Aand Bassociated with the inter-
face per unit interface area [67]:
c
AjB
ðTÞ ¼
H
AjB
T  S
AjB
O
AjB
ð19Þ
where H
AjB
ð¼ H
AþB
H
A
H
B
Þ and S
AjB
ð¼ S
AþB

S
A
S
B
Þ are the excess enthalpy and excess entropy of the
system due to the presence of the AjB interface area,O
AjB
:
i.e.the differences between the actual total enthalpy
(H
AþB
) and total entropy (S
AþB
) of the system and the sum
of the enthalpies (i.e.H
A
þH
B
) and entropies (S
A
þS
B
)
of the individual homogenous solids Aand Bin the absence
of the interface,i.e.if they were undisturbed by the divid-
ing AjBinterface [67].Compositional variations of the bulk
solids Aand 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 en-
ergy of an interface between two solids (denoted as SS in-
terface) is not possible by experiment (see what follows).
Indirect determination of the SS interface energy,c
AjB
,is
attempted by experiment by relating measured quantities
(e.g.adhesive forces,interfacial fracture,cleavage ener-
gies,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,
W
sep
AjB
,is defined as the reversible work,performed in a kind
of “Gedankenexperiment”,to separate the system at the
solid–solid AjB interface,thereby creating two free sur-
faces of the corresponding solids A and B,whereby plastic
and diffusional degrees of freedomupon separation are sup-
posed to be suppressed [84].The value of W
sep
AjB
is expressed
by the ideal Dupré equation,i.e.
W
sep
AjB
¼ c
S;unrelaxed
A
þc
S;unrelaxed
B
c
AjB
ð20Þ
where c
S;unrelaxed
A
and c
S;unrelaxed
B
denote the energies of the
respective “unrelaxed” A and B surfaces at infinite separa-
tion (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,c
AjB
,from the measured
strength of the SS interface is affected by numerous (also
interdependent) factors and side-effects,such as the geome-
try of the loading,the plastic and elastic properties of Aand
B,residual internal strains,defect formation and dislocation
movement (i.e.plastic flow),crack formation and propaga-
tion,the presence and size of flaws (e.g.interface rough-
ness,chemical impurities),diffusional processes for chemi-
cal equilibration (e.g.chemical interaction of the free
surfaces with the ambient;surface segregation) [78,80,
82–84].Therefore,the energy consumed in any conceiv-
able cleavage experiment generally substantially deviates
from (i.e.exceeds) the fundamental value of W
sep
AjB
accord-
ing to Eq.(20) [78,84,85],thereby invalidating reliable de-
termination of c
AjB
(supposing that accurate values for
c
S;unrelaxed
A
and c
S;unrelaxed
B
are available
3
).
L.P.H.Jeurgens et al.:Thermodynamics of reactions and phase transformations at interfaces and surfaces
1290 Int.J.Mat.Res.(formerly Z.Metallkd.) 100 (2009) 10
F
Feature
Fig.8.Surface energies,c
S;0
hMOi
,of the low-index crystallographic faces
of crystalline oxides with a rock-salt structure at T = 0 Kversus the en-
ergy term,E
lattice
hMOi
 V
0
hMOi
2=3
 N
A
1=3
.The indicated value of Ucor-
responds to the slope of the line fitted through the concerned data
points according to Eq.(18) in Section 3.2.2.[3].
3
Note that,for a crystalline solid of e.g.phase hAi,it holds that the “un-
relaxed” surface energy,c
S;unrelaxed
hAi
,representing the created solid sur-
face of hAi before its equilibration with the ambient,is different from
(i.e.generally higher than) the respective “relaxed” surface energy,
c
S
hAi
,which corresponds to the solid surface of hAi in equilibriumwith
the ambient.
On the contrary,it is much easier to obtain accurate ex-
perimental measures for the energy,c
hAijfBg
,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
c
S
hAi
c
hAijfBg
c
S
fBg
ð21Þ
where c
S
hAi
and c
S
fBg
denote the surfaces energies of crystal-
line hAi and liquid {B} in equilibrium with the ambient.
Thus obtained experimental values for solid–liquid (SL)
hAij{B} interface energies,c
hAijfBg
,are often considered
as approximate measures for the corresponding solid–solid
hAijhBi interface energies,c
hAijhBi
.
However,the energy of the interface between two crystal-
line 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 resid-
ual strain induced by misfit dislocations;see Section 4.3.).
Such structural energy contributions due to the lattice mis-
match between adjoining crystalline solids do not occur for
the corresponding solid–liquid hAij{B} interface (Sec-
tion 4.2.) and therefore it often holds that c
hAijfBg
< c
hAijhBi
[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 appli-
cation of (semi-)empirical formulations as derived from ex-
perimental data sets:cf.Refs.[3,21,22,38,84–90] and ref-
erences therein.Unfortunately,first-principle calculations,
but also the mentioned (semi-)empirical formulations,of SS
interface energies rely on detailed preknowledge of the inter-
face 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-re-
solved 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 de-
scriptions of solid–solid interfacial energies as a function
of,e.g.,phase composition and structure,temperature and
crystallographic orientation of the interface,which knowl-
edge is mandatory for the thermodynamic treatment pre-
sented in this paper,can only be readily and successfully
assessed by employing semi-empirical expressions,in par-
ticular those derived on the basis of the macroscopic atom
approach [1–3,40–46],as dealt with in the following sec-
tions.
4.1.Amorphous–amorphous interfaces
between (semi-)metals
Viscous flow in amorphous phases (considered as under-
cooled liquids) is relatively easy (as compared to crystalline
solids),because of their relatively large free volume and
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) en-
ergy contributions to the resultant interface energy,
c
fAgjfBg
,due to mismatch strain between {A} and {B} at
the {A}j{B} interface can be ignored.The resultant
{A}j{B} interface energy,c
fAgjfBg
,then only contains ex-
cess (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.electro-
static)) interactions of {A} and {B} across the {A}j{B} in-
terface [44,84,88,95,96]:
c
fAgjfBg
¼
H
fAgjfBg
T  S
fAgjfBg
O
fAgjfBg
¼ c
interaction
fAgjfBg
þc
entropy
fAgjfBg
ð22Þ
The interface enthalpy energy contribution,c
interaction
fAgjfBg
¼
H
interaction
fAgjfBg
=O
fAgjfBg
(per unit area interface),due to the in-
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)):
c
interaction
fAgjfBg
¼
1
2

f
fAg
 D
￿
H
1
fAg!fBg
O
fAg
þ
f
fBg
 D
￿
H
1
fBg!fAg
O
fBg
"#

D
￿
H
1
fAg!fBg
þD
￿
H
1
fBg!fAg
C
0
 ðV
2=3
fAg
þV
2=3
fBg
Þ
ð23aÞ
where D
￿
H
1
fAg!fBg
and D
￿
H
1
fBg!fAg
denote the partial en-
thalpies of dissolving one mole {A} in {B} and of one mole
{B} in {A},at infinite dissolution,respectively;O
fAg
and
O
fBg
correspond to the molar interface areas of {A} and
{B} (see Eq.(13) in Section 3.1.1.).
Values for D
￿
H
1
fAg!fBg
and D
￿
H
1
fBg!fAg
at the tempera-
ture concerned (and typically at 1 atm pressure) can easily
be extracted fromcorresponding 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
to define D
￿
H
1
fAg!fBg
and D
￿
H
1
fBg!fAg
in Eq.(23a) with re-
spect to A and B in the (undercooled) liquid state.However,
a direct comparison between calculated values for amor-
phous–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 addi-
tional superscript,“cr”:i.e.D
￿
H
1;cr
fAg!fBg
and D
￿
H
1;cr
fBg!fAg
.As
a result,the expression for the interface enthalpy energy con-
tribution for the {A}j{B} interface between amorphous {A}
and amorphous {B} becomes (cf.Eq.(23a))
c
interaction
fAgjfBg

D
￿
H
1;cr
fAg!fBg
þD
￿
H
1;cr
fBg!fAg
C
0
 ðV
2=3
fAg
þV
2=3
fBg
Þ
ð23bÞ
where D
￿
H
1;cr
fAg!fBg
ffi D
￿
H
1
fAg!fBg
þ DH
hBi!fBg
and D
￿
H
1;cr
fBg!fAg
ffi D
￿
H
1
fBg!fAg
þ DH
hAi!fAg
with DH
hBi!fBg
¼ ½DH
fBg
ðTÞ DH
hBi
ðTÞ and DH
hAi!fAg
¼
L.P.H.Jeurgens et al.:Thermodynamics of reactions and phase transformations at interfaces and surfaces
Int.J.Mat.Res.(formerly Z.Metallkd.) 100 (2009) 10 1291
F
Feature
½DH
fAg
ðTÞ DH
hAi
ðTÞ (at the temperature concerned and
at 1 atmpressure),respectively.For comprehensive discus-
sion on the necessity of explicit definition of the thermody-
namic reference state,see Refs.[43,44,46].
The interface entropy contribution,c
entropy
fAgjfBg
¼ T
S
fAgjfBg
O
fAgjfBg

(per unit area interface) predominantly
originates from the change in vibrational entropy of the in-
teracting {A} and {B} phases,as compared to the “bulk”
of these amorphous phases,at the {A}j{B} interface.As-
suming similar sizes of the atomic cells of the interacting
{A} and {B} atoms at the {A}j{B} interface (i.e.
V
2=3
fAg
 V
2=3
fBg
),an estimate of c
entropy
fAgjfBg
for the {A}j{B} in-
terface between two amorphous (or liquid) metals {A} and
{B} is obtained from[43,97]:
4
c
entropy
fAgjfBg
¼
1
1
2
 O
fAg
þO
fBg
 
 T  f
fAg
 3R
 ln
ffiffiffiffiffiffiffiffiffiffi
H
fAg
q

ffiffiffiffiffiffiffiffiffiffi
H
fBg
q
1
2
 H
fAg
þH
fBg
 
þDH
fAgjfBg
2
4
3
5

6R  T
C
0
 V
2=3
fAg
þV
2=3
fBg


 ln
ffiffiffiffiffiffiffiffiffiffi
H
fAg
q

ffiffiffiffiffiffiffiffiffiffi
H
fBg
q
1
2
 H
fAg
þH
fBg
 
þDH
fAgjfBg
2
4
3
5
ð24aÞ
Here H
fAg
and H
fBg
denote the Debye temperatures of {A}
and {B};DH
fAgjfBg
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 corre-
sponding bulk phases of {A} and {B},which can be ex-
pressed by [43,97]
DH
fAgjfBg
¼ 34:1  10
3
 ðDH
fABg
=RÞ ð24bÞ
where DH
fABg
ðTÞ denotes the temperature-dependent (bulk)
enthalpy of formation of the {AB} solid solution of equi-
atomic composition.
Substitution of Eqs.(23),(24a) and (24b) in Eq.(22) fi-
nally leads to Eq.(25) (see bottomof 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 re-
sultant hAij{B} interface energy,c
hAijfBg
,can then be ex-
pressed as the resultant of three additive interfacial energy
contributions [1–3,43–45]:
c
hAijfBg
¼
H
hAijfBg
T  S
hAijfBg
O
hAijfBg
¼ c
enthalpy
hAijfBg
þc
interaction
hAijfBg
þc
entropy
hAijfBg
ð26Þ
The enthalpy contribution,c
enthalpy
hAijfBg
,arises from the rela-
tive 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 con-
tribution,c
interaction
hAijfBg
,results fromthe physical and chemical
interactions of hAi and {B} across the hAij{B} interface
(cf.Section 4.1.).Furthermore it is assumed that the vibra-
tional 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 in-
teracting 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
contribution,c
enthalpy
hAijfBg
(cf.Eq.(26)),due to the relative in-
crease in enthalpy of crystalline hAi at the interface (as
compared to bulk crystalline hAi),is approximated by [1,
2,43–45,98]
c
enthalpy
hAijfBg
¼
f
hAi
 DH
hAi!fAg
O
hAi

H
fAg
ðTÞ H
hAi
ðTÞ
C
0
 V
2=3
hAi
ð27Þ
with f
hAi
%0.35 (Section 3.1.2.).
The interfacial interaction contribution,c
interaction
hAijfBg
,is given
by(cf.Eq.(23) inSection 4.1.withf
hii
ffi f
fig
for i = A,B[44])
c
interaction
hAijfBg
¼
1
2

f
hAi
 D
￿
H
1;cr
fAg!fBg
O
hAi
þ
f
fBg
 D
￿
H
1;cr
fBg!fAg
O
fBg
"#

1
2

D
￿
H
1;cr
fAg!fBg
C
0
 V
2=3
hAi
þ
D
￿
H
1;cr
fBg!fAg
C
0
 V
2=3
fBg
2
4
3
5
ð28Þ
with D
￿
H
1;cr
fAg!fBg
ffi D
￿
H
1
fAg!fBg
þ DH
hBi!fBg
and
D
￿
H
1;cr
fBg!fAg
ffi D
￿
H
1
fBg!fAg
þ DH
hAi!fAg
(see Section 4.1.)
and where it has been recognized that the layer of hAi adja-
cent to {B} can thermodynamically be approximated by
{A} [1,2,43–45,98].
L.P.H.Jeurgens et al.:Thermodynamics of reactions and phase transformations at interfaces and surfaces
1292 Int.J.Mat.Res.(formerly Z.Metallkd.) 100 (2009) 10
F
Feature
c
fAgjfBg
¼
D
￿
H
1;cr
fAg!fBg
þD
￿
H
1;cr
fBg!fAg
6R  T  ln
ffiffiffiffiffiffiffiffiffiffi
H
fAg
q

ffiffiffiffiffiffiffiffiffiffi
H
fBg
q
1
2
 ðH
fAg
þH
fBg
Þ þ34:1  10
3
 ðDH
fABg
=RÞ
2
4
3
5
C
0
 V
2=3
fAg
þV
2=3
fBg


ð25Þ
4
Similar (approximate) expressions for the interfacial vibrational en-
tropy contribution can be derived for {A}|hABi or hAi|{AB}interfaces
on the basis of the guidelines provided by Ref.[43].
The interfacial entropy contribution,c
entropy
hAijfBg
,due to the
decrease in configurational entropy of the amorphous phase
{B} at the hAij{B} interface,is estimated by [1,99,100]:
c
entropy
hAijfBg
¼ T 
0:678  R
O
fBg

0:678  R  T
f
fBg
 C
0
 V
2=3
fBg
ð29Þ
with f
fBg

1
3
.
Substitution of Eqs.(27–29) into Eq.(26) yields
c
hAijfBg

½H
fAg
ðTÞ H
hAi
ðTÞ þ
1
2
 D
￿
H
1;cr
fAg!fBg
C
0
 V
2=3
hAi
þ
0:678  R  T  ð f
fBg
Þ
1
þ
1
2
 D
￿
H
1;cr
fBg!fAg
C
0
 V
2=3
fBg
ð30Þ
4.2.2.Crystalline–amorphous interfaces
between metals and oxides
For the interface between a crystalline metal,hMi,and its
amorphous oxide,fM
x
O
y
g,the enthalpy contribution,
c
enthalpy
hMijfM
x
O
y
g
(cf.Eq.(26)),due to the enthalpy increase of
crystalline hMi at the hMijfM
x
O
y
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])
c
enthalpy
hMijfM
x
O
y
g
¼
f
hMi
 DH
hMi!fMg
O
hMi
¼
f
hMi
 H
fMg
ðTÞ H
hMi
ðTÞ
 
O
hMi
ð31Þ
with f
hMi
%0.35 (Section 3.1.2.).The molar interface area,
O
hMi
,of metal atoms of hMi at the hMijfM
x
O
y
g interface
follows fromthe area enclosed by one mole of metal atoms
in the crystallographic plane of hMi at the hMijfM
x
O
y
g in-
terface plane (alternatively,an “average-crystal-plane” val-
ue of O
hMi
is obtained from O
hMi
ffi f
hMi
 C
0
 V
2=3
hMi
;see
Eq.(15) in Section 3.1.2.).
The interfacial interaction contribution,c
interaction
hMijfM
x
O
y
g
,due
to the interactions of hMi and fM
x
O
y
g across the
hMijfM
x
O
y
g interface,is given by (see Section 4.1.and
Ref.[2])
c
interaction
hMijfM
x
O
y
g
¼
f
fOg
 D
￿
H
1
O!hMi
O
fOg
ð32Þ
where D
￿
H
1
O!hMi
denotes the partial enthalpy of dissolving
one mole O atoms at infinite dissolution in (solid) crystal-
line hMi and O
fOg
is the interface area enclosed by one
mole of oxygen ions in the amorphous oxide at the
hMijfM
x
O
y
g interface (f
fOg

1
3
;see Section 3.1.1.).Evi-
dently,for metal j oxide interfaces hM
I
ijfM
II
x
O
y
g where
M
I
6¼ M
II
,an additional (excess) interaction energy term,
f
fM
II
g
 D
￿
H
1;cr
fM
II
g!fM
I
g
=O
fM
II
g
,(cf.Eq.(28) and related dis-
cussion) should to be added to the interfacial interaction
contribution according to Eq.(32),to account for the physi-
cal and chemical interactions between dissimilar metal
atoms across the hM
I
ijfM
II
x
O
y
g interface:for details,see
Refs.[2,40,102].
The entropy contribution,c
entropy
hMijfM
x
O
y
g
,due to the decrease
in configurational entropy of the amorphous phase fM
x
O
y
g
at the hMijfM
x
O
y
g interface,can be estimated fromthe en-
tropy difference between bulk amorphous fM
x
O
y
g and the
corresponding bulk crystalline oxide phase,hM
x
O
y
i,i.e.
S
hM
x
O
y
i
 S
fM
x
O
y
g
[2].The entropy contribution per unit area
interface then becomes:
c
entropy
hMijfM
x
O
y
g
¼ T 
S
hM
x
O
y
i
ðTÞ S
fM
x
O
y
g
ðTÞ
y  O
fOg
ð33Þ
where y equals the number of O ions per stoichiometric
M
x
O
y
molecule (and O
fOg
has been defined below
Eq.(32)).Substitution of Eqs.(31–33) in Eq.(26) gives
Eq.(34) (see bottomof page).
For those metal–oxide systems for which the value of
D
￿
H
1
O!hMi
is unknown,a value can be estimated from the
established empirical relation between D
￿
H
1
O!hMi
(in [J (mole
O)
– 1
]) and the corresponding enthalpy of formation,
DH
f
hM
x
O
y
i
(in [J (mole M
x
O
y
)
– 1
]) of the crystalline oxide,
hM
x
O
y
i,out of its stable elements [2,3]:
D
￿
H
1
O!hMi
ffi 1:2  y
1
 DH
f
hM
x
O
y
i
ðTÞ þ1  10
5
ð35Þ
For most metal–oxide systems,the metal–oxygen bond
formation [and thus the value of D
￿
H
1
O!hMi
employed in
Eq.(34)] is strongly exothermic [3,103].Consequently,the
resultant crystalline–amorphous interface energy,c
hMijfM
x
O
y
g
,
is generally predominated by the relatively large negative
metal–oxygen interaction energy,c
interaction
hMijfM
x
O
y
g
[2,3] (note:this
also holds for the corresponding crystalline–crystalline
interface energy,c
hMijhM
x
O
y
i
[42];see Section 4.3.2.).For
example,the positive sum of the enthalpy (c
enthalpy
hMijfM
x
O
y
g
) and
entropy (c
entropy
hMijfM
x
O
y
g
) contributions is typically smaller than
+0.3 ± 0.2 J m
–2
,whereas the corresponding negative in-
teraction contribution,c
interaction
hMijfM
x
O
y
g
,is in the range of
–4.5 J m
– 2
to –1.0 J m
–2
(e.g.c
interaction
hMgijfMgOg
ffi c
interaction
hAlijfAl
2
O
3
g

c
interaction
hZrijfZrO
2
g
*–4.5 ± 0.2 J m
–2
;c
interaction
hCrijfCr
2
O
3
g
,c
interaction
hSiijfSiO
2
g
,
c
interaction
hNiijfNiOg
and c
interaction
hFeijfFe
2
O
3
g
are in between –2 and –1 J m
–2
;
as an exception c
interaction
hCuijfCuO
2
g
*–0.2 ± 0.1 J m
–2
) [2,3,41,
104].This implies that the lowest value of c
hMijfM
x
O
y
g
,and
thus the most stable (i.e.thermodynamically-preferred)
hMijfM
x
O
y
g interface,is generally achieved by maximiz-
ing the density of metal–oxygen bonds across the
hMijfM
x
O
y
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 hMijfM
x
O
y
g
interface,in accordance with recent experimental observa-
tions [42].Agood approximate for the molar interface area,
O
fOg
,enclosed by one mole of oxygen atoms in the amor-
phous oxide at the hMijfM
x
O
y
g interface (see Eq.(34)),is
L.P.H.Jeurgens et al.:Thermodynamics of reactions and phase transformations at interfaces and surfaces
Int.J.Mat.Res.(formerly Z.Metallkd.) 100 (2009) 10 1293
F
Feature
c
hMijfM
x
O
y
g
¼
f
hMi
 ½H
fMg
ðTÞ H
hMi
ðTÞ
O
hMi
þ
f
fOg
 D
￿
H
1
O!hMi
T  y
1
 ½S
hM
x
O
y
i
ðTÞ S
fM
x
O
y
g
ðTÞ
O
fOg
ð34Þ
therefore obtained from such area enclosed by one mole of
Oions in the most densely-packed plane of the correspond-
ing (unstrained) crystalline oxide phase,hM
x
O
y
i,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,c
hAijhBi
,of the hAijhBi interface be-
tween 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 en-
tropy contributions due to the physical and chemical inter-
actions between hAi and hBi across the interface (as already
introduced in Sections 4.1.and 4.2.),whereas the structural
term (further designated as c
mismatch
hAijhBi
) originates from the
mismatch between the adjoining crystal structures of hAi
and hBi at the hAijhBi interface.For a fully coherent inter-
face,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 re-
gime”,generally only occurs for small initial lattice mis-
matches 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 re-
ferred to as the “mixed regime”),the residual homogeneous
strains in hAi and/or hBi,c
strain
hAijhBi
,can thought to be super-
imposed on the periodic,inhomogeneous strain field,
c
dislocation
hAijhBi
,resulting fromthe sumof strain fields associated
with each of the misfit dislocations at the semi-coherent
hAijhBi interface [41,106],i.e.
c
hAijhBi
¼ c
interaction
hAijhBi
þc
entropy
hAijhBi
þc
mismatch
hAijhBi
¼ c
interaction
hAijhBi
þc
entropy
hAijfBg
þc
strain
hAijhBi
þc
dislocation
hAijhBi
ð36Þ
If all residual mismatch strains in hAi and hBi are fully re-
laxed by the generation of misfit dislocations at the hAijhBi
interface,the (fully) “plastic regime” has been entered
(i.e.c
strain
hAijhBi
¼ 0 and c
mismatch
hAijhBi
¼ c
dislocation
hAijhBi
).
For the interface energies between two crystalline solids,
as a rule of thumb,the interfacial interaction energy contribu-
tion,c
interaction
hAijhBi
,to the resultant interface energy,is much lar-
ger (i.e.less negative or even slightly positive) for interfaces
between (semi-)metals (with c
interaction
hAijhBi
values in the range of
–0.5 J m
– 2
to +1.0 J m
–2
) than the similar contribution for
interfaces between (semi-)metals and oxides (with negative
c
interaction
hMijhM
x
O
y
i
values in the range of about –4.5 J m
– 2
to
–1.0 J m
– 2
;cf.Section 4.2.2.).
Similar to hMijfM
x
O
y
g interfaces (see Section 4.2.2.),the
contribution of c
interaction
hMijhM
x
O
y
i
to the resultant interface energy,
c
hMijhM
x
O
y
i
,is generally dominant for hMijhM
x
O
y
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 hMijhM
x
O
y
i interface (see e.g.Refs.[3,41,42,
56] and Section 4.3.2.),often plays a crucial role for the in-
terface thermodynamics of oxide phases in contact with met-
als [42,56].
On the contrary,the crystallographic orientation relation-
ship (OR) between adjoining crystalline (semi-)metals can
often,to a first approximation,be disregarded in thermody-
namic model descriptions of phase transformations in
(small-dimensional) (semi-)metal systems (i.e.average-crys-
tal plane values can be adopted,instead of orientation-depen-
dent 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 inter-
action (c
interaction
hAijhBi
) and entropy (c
entropy
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.)
c
interaction
hAijhBi
¼
1
2

f
hAi
 D
￿
H
1;cr
hAi!hBi
O
hAi
þ
f
hBi
 D
￿
H
1;cr
hBi!hAi
O
hBi
"#

D
￿
H
1;cr
hAi!hBi
þD
￿
H
1;cr
hBi!hAi
C
0
 V
2=3
hAi
þV
2=3
hBi


ð37Þ
and
c
entropy
hAijhBi

6R  T
C
0
 V
2=3
hAi
þV
2=3
hBi


ð38Þ
 ln
ffiffiffiffiffiffiffiffiffi
H
hAi
q

ffiffiffiffiffiffiffiffiffi
H
hBi
q
1
2
 H
hAi
þH
hBi
 
þ34:1  10
3
 DH
hABi
=R


2
4
3
5
where DH
hABi
is defined as the enthalpy of formation of the
<AB> solid solution of equiatomic composition;D
￿
H
1;cr
hAi!hBi
and D
￿
H
1;cr
hBi!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 cor-
L.P.H.Jeurgens et al.:Thermodynamics of reactions and phase transformations at interfaces and surfaces
1294 Int.J.Mat.Res.(formerly Z.Metallkd.) 100 (2009) 10
F
Feature
c
hAijhBi
¼
D
￿
H
1;cr
hAi!hBi
þD
￿
H
1;cr
hBi!hAi
6R  T  ln
ffiffiffiffiffiffiffiffiffi
H
hAi
q

ffiffiffiffiffiffiffiffiffi
H
hBi
q
1
2
 ðH
hAi
þH
hBi
Þ þ34:1  10
3
 ðDH
hABi
=RÞ
2
4
3
5
C
0
 V
2=3
hAi
þV
2=3
hBi


þ
c
S
hAi
ðTÞ þc
S
hBi
ðTÞ
6
ð40Þ
responding crystalline metals,which empirically equals
[45,107] about one third of the surface energy of the re-
spective crystalline metal in contact with vacuum (as de-
fined in Section 3.1.2.),i.e.
c
mismatch
hAijhBi
¼
1
3

c
S
hAi
ðTÞ þc
S
hBi
ðTÞ
2
"#
¼
c
S
hAi
ðTÞ þc
S
hBi
ðTÞ
6
ð39Þ
Substitution of Eqs.(37–39) in Eq.(36) gives Eq.(40) (see
bottomof previous page).
4.3.2.Crystalline–crystalline interfaces
between metals and oxides
For (semi-)coherent hMijhM
x
O
y
i interfaces between a crys-
talline metal,hMi,and its crystalline oxide,hM
x
O
y
i,the ex-
cess entropy contribution to c
hMijhM
x
O
y
i
is negligibly small
(as compared to c
interaction
hMijhM
x
O
y
i
and c
mismatch
hMijhM
x
O
y
i
;see introductory
part of Section 4.3.) and is therefore neglected.In the follow-
ing,the interfacial interaction,strain,and dislocation contri-
butions to c
hMijhM
x
O
y
i
(see Eq.(36)) will be evaluated for the
case of an overgrowth of an ultra-thin oxide film (of,say,
h
hM
x
O
y
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
hM
x
O
y
i on hMi (i.e.in the fully elastic regime;see intro-
duction of Section 4.3.),all lattice mismatch is accommo-
dated fully elastically by the thin epitaxial oxide film:i.e.
c
dislocation
hMijhM
x
O
y
i
= 0.The homogeneous,normal strains,e
11
and
e
22
,in the oxide film in mutually perpendicular directions
1 and 2 along the coherent hMijhM
x
O
y
i interface plane,
are then given by the initial lattice mismatch values f
1
and
f
2
along the corresponding directions 1 and 2 within the in-
terface plane,respectively (as governed by the OR between
hMi and hM
x
O
y
i):
e
ii
¼ f
i
¼
i
a
hMi

i
a
hM
x
O
y
i
i
a
hM
x
O
y
i
ði ¼ 1;2Þ ð41Þ
where
i
a
hMi
and
i
a
hM
x
O
y
i
denote values of unstrained lattice
spacings of hMi and hM
x
O
y
i along the corresponding per-
pendicular directions 1 and 2,respectively,within the inter-
face plane.
However,with increasing oxide film thickness,h
hM
x
O
y
i
,
as well as at the initial stage of crystalline oxide formation
for hMijhM
x
O
y
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 hMijhM
x
O
y
i
interface.With increasing density of misfit dislocations
at the interface,the strain contribution,c
strain
hMijhM
x
O
y
i
,de-
creases,whereas the dislocation contribution,c
dislocation
hMijhM
x
O
y
i
,
increases [41,106].
In the above sketched situation (i.e.in the mixed regime),
at a given oxide thickness,h
hM
x
O
y
i
,and growth temperature,
T,the interfacial interaction energy contribution,c
interaction
hMijhM
x
O
y
i
,
then is given by [41] (cf.Eq.(32) in Section 4.2.2.)
c
interaction
hMijhM
x
O
y
i
¼
f
hOi
 D
￿
H
1
O!hMi
O
hOi
 1 þ
￿
e
11
ð Þ  1 þ
￿
e
22
ð Þ ð42Þ
where the molar interface area,O
hOi
,pertains to the “hy-
pothetically” unstrained crystal plane of hM
x
O
y
i at the
hMijhM
x
O
y
i interface (with f
hOi
%0.35).The additional
term ð1 þ
￿
e
11
Þð1 þ
￿
e
22
Þ in Eq.(42) is introduced to correct
for the area difference between the strained and unstrained
crystalline oxide film,where
￿
e
11
and
￿
e
22
denote the (thick-
ness-dependent) residual,homogeneous,normal strains in
the oxide in the mutually perpendicular directions 1 and 2
along the hMijhM
x
O
y
i interface plane at the growth tem-
perature,T.The residual strains,
￿
e
ii
,are related to the corre-
sponding residual lattice spacings,
i
￿
a
hM
x
O
y
i
,of the hM
x
O
y
i
film along the perpendicular directions 1 and 2 within the
interface plane,by
￿
e
ii
¼
i
￿
a
hM
x
O
y
i

i
a
hM
x
O
y
i
i
a
hM
x
O
y
i
ði ¼ 1;2Þ ð43Þ
The strain energy contribution,c
strain
hMijhM
x
O
y
i
,due to the state
of residual homogeneous strain in the crystalline oxide film
at h
hM
x
O
y
i
and T,is obtained from[41]
c
strain
hMijhM
x
O
y
i
¼ h
hM
x
O
y
i

￿
r
ij

￿
e
ij




¼ h
hM
x
O
y
i
 C
ijkl

￿
e
ij

￿
e
kl




ði;j;k;l ¼ 1;2;3Þ ð44Þ
where
￿
r
ij
is the stress tensor,C
ijkl
is the fourth-rank stiffness
tensor,and
￿
e
ij
is the residual homogeneous strain tensor of
hM
x
O
y
i (with direction 3 perpendicular to the interface plane).
Finally,the dislocation energy contribution,c
dislocation
hMijhM
x
O
y
i
,
at h
hMO
x
i
and T,is given by:
c
dislocation
hMijhM
x
O
y
i
¼
1
c
dislocation
hMijhM
x
O
y
i






þ
2
c
dislocation
hMijhM
x
O
y
i






ð45Þ
with
1
c
dislocation
hMijhM
x
O
y
i
and
2
c
dislocation
hMijhM
x
O
y
i
as the energies of the two
mutually perpendicular,regularly spaced arrays of misfit
dislocations with Burgers vectors parallel to the two per-
pendicular directions 1 and 2 in hM
x
O
y
i [41].
Different approaches for crystalline misfit accommoda-
tion at an interface (as reported in the literature) have been
compared and evaluated in Ref.[41] to asses the misfit-en-
ergy contribution
i
c
dislocation
hMijhM
x
O
y
i
.It was found that the so-called
“First Approximation” approach (APPR) of Frank and van
der Merwe (for details,see Refs.[41,106]) has the greatest
overall accuracy for the estimation of
i
c
dislocation
hMijhM
x
O
y
i
ðh
hM
x
O
y
i
;TÞ for a wide range of initial lattice-mismatch va-
lues in both the monolayer and nanometer thickness re-
gimes (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
applicable for calculation of
i
c
dislocation
hMijhM
x
O
y
i
ðh
hM
x
O
y
i
;TÞ.
Substitution of Eqs.(42),(44),and (45) in Eq.(36) then
gives (assuming c
entropy
hMijhM
x
O
y
i
&0;see above):
c
hMijhM
x
O
y
i
¼
f
hOi
 D
￿
H
1
O!hMi
O
hOi
 1 þ
￿
e
11
ð Þ  1 þ
￿
e
22
ð Þ
þh
hM
x
O
y
i
 C
ijkl

￿
e
ij

￿
e
kl




þ
1
c
dislocation
hMijhM
x
O
y
i






þ
2
c
dislocation
hMijhM
x
O
y
i






ð46Þ
Due to the thickness dependence of both
￿
e
ii
and c
dislocation
hMijhM
x
O
y
i
(and thus of c
hMijhM
x
O
y
i
;see Eqs.(43–46)),the calculation
L.P.H.Jeurgens et al.:Thermodynamics of reactions and phase transformations at interfaces and surfaces
Int.J.Mat.Res.(formerly Z.Metallkd.) 100 (2009) 10 1295
F
Feature
of c
hMijhM
x
O
y
i
as a function of h
hM
x
O
y
i
and T can only be per-
formed numerically.To this end,the resultant interface en-
ergy,c
hMijhM
x
O
y
i
,according to Eq.(46) is minimized with
respect to the residual homogeneous strain in the film for a
given oxide-film thickness,h
hM
x
O
y
i
,and growth tempera-
ture,T,by introducing
1
￿
a
hM
x
O
y
i
and
2
￿
a
hM
x
O
y
i
as the only fit
parameters,values of which are obtained by requiring:
qc
hMijhM
x
O
y
i
q
￿
e
ij
¼ 0 ð47Þ
Appropriate literature references providing elastic con-
stants of metals and crystalline oxides,as required for the
calculation of c
hMijhM
x
O
y
i
,have been listed in Refs.[3,41].
The “unstrained” molar interface area,O
hOi
,(see Eq.(46)),
as well as the “unstrained” lattice parameters,
i
a
hM
x
O
y
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 hM
x
O
y
i;one should generally refrain
from adopting an average-crystal-plane value for O
hOi
,
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
c
interaction
hMijhM
x
O
y
i
and c
interaction
hMijfM
x
O
y
g
,as calculated using Eqs.(42)
and (32),arise only from differences in the values of O
hOi
and O
fOg
and the presence of residual strains,
￿
e
11
and
￿
e
22
,
in the crystalline oxide overgrowth.
Thus obtained values for the interfacial energy contributions
due to strain (c
strain
hMijhM
x
O
y
i
),misfit dislocations (c
dislocation
hMijhM
x
O
y
i
) and
chemical interaction (c
interaction
hMijhM
x
O
y
i
),as well as the value of the
resultant hMijhM
x
O
y
i interface energy (c
hMijhM
x
O
y
i
),have
been plotted in Fig.9 as function of the oxide-film thick-
ness (h
hM
x
O
y
i
) for the hNif111gijhNiOf100gi,the
hZrf0001gijhZrO
2
f1
￿
11gi and the hTif10
￿
10gijhTiO
2
f100gi
interface.
It follows that,because of the lowinitial lattice mismatch
of only +3%and –0.3%along two mutually perpendicular
directions parallel to the hTif10
￿
10gijhTiO
2
f100gi interface
plane (see Eq.(41)),all mismatch strain is accommodated
fully elastically by the initial hTiO
2
f100gi overgrowth on
hTif10
￿
10gi (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 there-
fore increases linearly with increasing oxide-filmthickness
until a first array of misfit dislocations is introduced in the
overgrowth at the metal/oxide interface along the highest
mismatch direction for h
hTiO
2
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 lat-
tice 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 introduc-
tion of misfit dislocations at the hNif111gijhNiOf100gi in-
terface already at the onset of growth along the high-mis-
match direction and,subsequently,also along the low-
mismatch 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,
c
interaction
hNif111gijhNiOf100gi
,due to the associated increase of the den-
sity of O–Ni bonds across the interface (Fig.9c).
L.P.H.Jeurgens et al.:Thermodynamics of reactions and phase transformations at interfaces and surfaces
1296 Int.J.Mat.Res.(formerly Z.Metallkd.) 100 (2009) 10
F
Feature
(a) (b)
(c) (d)
Fig.9.(a) Strain energy contribution
(c
strain
hMijhM
x
O
y
i
),(b) misfit dislocation energy
contribution (c
dislocation
hMijhM
x
O
y
i
),(c) interaction en-
ergy contribution (c
interaction
hMijhM
x
O
y
i
) and (d) resul-
tant interfacial energy of the hMijhM
x
O
y
i in-
terface (c
hMijhM
x
O
y
i
) as function of the oxide-
film thickness (h
hM
x
O
y
i
) for the hNif111gij
hNiOf100gi,the hZrf0001gijhZrO
2
f1
￿
11gi
and the hTif10
￿
10gijhTiO
2
f100gi interface
[3] (For details,see Section 4.3.2.).
Analogously,if an initially compressive growth strain re-
sides in the oxide overgrowth,as for hZrO
2
f111gi on
hZrf0001gi (with a corresponding near-isotropic initial lat-
tice 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 con-
tribution (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 over-
growths (for metal–oxide systems with similar metal–oxy-
gen 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 neg-
ative.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 con-
tribution,c
interaction
hMijhM
x
O
y
i
(i.e.a more negative value of D
￿
H
1
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 nano-
technologies,such as tunnel junctions [23–25],gas sensors
[28,29],model catalysts [30,31],and (thin) diffusion bar-
riers for corrosion resistance [26,27].The microstructure
of these oxides often differ from those known and as pre-
dicted 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 en-
ergy of the amorphous oxide phase (as compared to that of
the competing crystalline oxide phase) can be overcompen-
sated by its lower sumof surface and interface energies (cf.
Refs.[3,108] and references therein;see also Section 2.2.).
Only if the thickness of the amorphous oxide filmexceeds 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,initi-
al 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 oxide-
film 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 ac-
cording to “bulk” thermodynamics,can be thermodynami-
cally preferred by their relatively low surface energies:
e.g.,c-Al
2
O
3
instead of a-Al
2
O
3
[73],c-Y
2
O
3
instead of
a-Y
2
O
3
[109] or tetragonal ZrO
2
instead of monoclinic
ZrO
2
[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].
Also,as predicted theoretically and found experimen-
tally [42],an unexpected crystallographic orientation rela-
tionship (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 for-
mation of the stable crystalline bulk modification.
Obviously,fundamental and comprehensive knowledge
on the thermodynamics of these nano-sized oxide micro-
structures is of utmost importance for the aforementioned
nano-technologies:one strives for either a stable amorphous
oxide phase or a stable coherent,single-crystalline (tem-
plate) oxide phase,because of the absence of grain bound-
aries in both oxide-filmmodifications;such grain boundaries
would act as paths for fast ionic or electronic migration,
thereby deteriorating material properties such as the electri-
cal 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-SiO
2
,a-Al
2
O
3
,(a-HfO
2
)
x
(a-Al
2
O
3
)
1–x
,because of
their uniformthickness and specific microstructure (no grain
boundaries,moderate bond flexibility,large free volume,
negligible growth strain) and related properties (e.g.,passi-
vating oxide-film growth kinetics,low leakage current,high
dielectric constant,high corrosion resistance) [57,115,116].
In the following two typical cases of theoretical predic-
tion and experimental verification of such nano-sized oxide
microstructures are presented.The first example (Sec-
tion 5.1.) deals with the thermodynamic stability of ultra-
thin amorphous oxide overgrowths on their metal substrates
L.P.H.Jeurgens et al.:Thermodynamics of reactions and phase transformations at interfaces and surfaces
Int.J.Mat.Res.(formerly Z.Metallkd.) 100 (2009) 10 1297
F
Feature
Fig.10.HRTEM micrograph and corresponding LEED pattern (at
100 eV) of the amorphous fAl
2
O
3
g overgrowth on hAlf111gi after
oxidation for t = 6000 s at T = 373 K and p
O
2
= 1∙ 10
– 4
Pa (and subse-
quent 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].
(as compared to the competing crystalline oxide).The sec-
ond 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 fAl
2
O
3
g
overgrowth on hAlf111gi after thermal oxidation for
t = 6000 s at T = 373 K is shown in Figure 10.All Al
2
O
3
films grown on hAlf111gi by thermal oxidation at
T £ 450 K are amorphous,as determined by LEED and
HRTEM,and have limiting,uniform thicknesses,h
fAl
2
O
3
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
fAl
2
O
3
g overgrowths comply well with the predicted critical
thickness of h
crit
fAl
2
O
3
g
= 0.7 ± 0.1 nm(*3–4 oxide monolayers
with 1 ML*0.22 nm) up to which an amorphous fAl
2
O
3
g
film is thermodynamically preferred on the hAlf111gi sub-
strate (for T = 350–900 K;compare Figs.11 and 12a).As
follows from the thermodynamic model calculations accord-
ing to the procedure outlined in Section 2.2.,the fAl
2
O
3
g
overgrowth on hAlf111gi is thermodynamically preferred
with respect to the competing hc-Al
2
O
3
i overgrowth due to
the slightly lower energy of the hAlð111ÞijfAl
2
O
3
g interface
(because the corresponding hc-Al
2
O
3
i overgrowth is tensilely
strained [3];see Section 4.3.2.) in combination with a rela-
tively small bulk energy difference between fAl
2
O
3
g and
hc-Al
2
O
3
i.If a high activation energy barrier exists for the
corresponding amorphous-to-crystalline transition,the initial
fAl
2
O
3
g overgrowth may be maintained for oxide-filmthick-
nesses beyond h
crit
fAl
2
O
3
g
.
Thermal oxidation of bare hAlf111gi substrates at more
elevated temperatures T = 475 K instead results in the
formation of (epitaxial) crystalline hc(-like)-Al
2
O
3
i films
beyond the critical thickness,h
crit
fAl
2
O
3
g
= 0.7 ± 0.1 nm
(Figs.11 and 12a).An HRTEM micrograph of a corre-
sponding epitaxial hc(-like)-Al
2
O
3
i 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 var-
ious 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 tem-
perature in Fig.14.Exemplary dependencies of the calcu-
lated critical oxide-film thicknesses on the metal substrate
orientation have been presented in Fig.12 for oxide over-
growths on different low-index crystallographic surfaces
of hAli,hCri,and hZri [3,41].
The strikingly high stability of the fSiO
2
g overgrowth on
hSif111gi (i.e.h
crit
fSiO
2
g
> 40 nm[3]) is due to the exception-
L.P.H.Jeurgens et al.:Thermodynamics of reactions and phase transformations at interfaces and surfaces
1298 Int.J.Mat.Res.(formerly Z.Metallkd.) 100 (2009) 10
F
Feature
Fig.11.Recorded oxide-film growth curves [15] for the thermal oxi-
dation of bare hAlf111gi and hAlf100gi substrates at 350 K,450 K,
and 550 K (at p
O
2
= 1∙ 10
–4
Pa).The presented oxide-film growth
curves were obtained by fitting experimental growth curves,as ob-
tained by real-time in-situ spectroscopic ellipsometry (RISE),with the-
oretical growth curves as calculated on the basis of the coupled cur-
rents 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,
h
crit
fAl
2
O
3
g
*0.7 ± 0.1 nm,up to which an amorphous fAl
2
O
3
g over-
growth (in instead of the corresponding crystalline hc-Al
2
O
3
i over-
growth) is predicted to be thermodynamically preferred on the parent
hAlf111gi and hAlf100gi substrates [42,56].
(a)
(b)
(c)
Fig.12.Calculated critical thickness,h
crit
fM
x
O
y
g
,up to which an amor-
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
growth temperature (T).The values of h
crit
fM
x
O
y
g
were determined by
solving h
fM
x
O
y
g
according to Eq.(8) for DG
cell
ðh
fM
x
O
y
g
;TÞ = 0.For de-
tails,see Sections 2.2,3.2.,and 4.
ally small bulk Gibbs energy difference between amor-
phous and crystalline SiO
2
in combination with the consid-
erably lower surface energy of amorphous fSiO
2
g (as com-
pared to hSiO
2
i;see also Fig.6c) [3,55].Amorphous
fSiO
2
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 correspond-
ing amorphous-to-crystalline transition.
For oxide overgrowths on hZrf0001gi and hTif0001gi,in
spite of the relatively low surface and interfacial energy for
the amorphous oxide-film configuration,the relatively large
difference in bulk energy between the amorphous and crys-
talline 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 ther-
modynamically stable,amorphous oxide film on these met-
al 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 dif-
ferences between amorphous and crystalline NiO are too
small to compensate the corresponding (positive) bulk en-
ergy 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.
h
crit
fM
x
O
y
g
< 0),in accordance with the limited number of ex-
perimental observations reported in the literature [117–
123].In these cases the (negative) sumof the surface and in-
terfacial energy differences of the amorphous and crystalline
oxide overgrowths are too small to overcompensate the cor-
responding (positive) bulk energy difference.
The dependence of h
crit
fM
x
O
y
g
on the metal substrate orienta-
tion (Fig.12) is mainly determined by differences in hM
x
O
y
i
surface energy and M–O bond density across the
hMijhM
x
O
y
i interface for the differently oriented crystalline
oxide overgrowths (as imposed by the OR between the crys-
talline oxide overgrowth and the parent metal substrate) [3].
5.2.Thermodynamic stability
of high-mismatch crystalline oxide films
In-situ LEED and ex-situ HRTEM analysis [42] indicate
the existence of a OR between the hc(-like)-Al
2
O
3
i over-
growth and the hAlf111gi substrate according to:
hAlð111Þ½110ijjhc-Al
2
O
3
ð111Þ½110i,which is the expected
OR with lowest possible mismatch (of about +2.0% at
T = 300 K) between hAlf111gi and hc-Al
2
O
3
i.
As for the thermal oxidation of bare hAlf111gi substrates,
an overall stoichiometric Al
2
O
3
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 oxi-
dation temperatures T < 450 K,the oxide films were found to
be amorphous.For T ‡ 450 K,a crystalline hc(-like)-Al
2
O
3
i
overgrowth develops on the hAlf100gi substrate (beyond an
experimentally verified critical oxide-filmthickness of about
0.45 ± 0.15 nm [56]) with a OR relationship according to
[42]:hAlð100Þ½011ijjhc-Al
2
O
3
ð111Þ½01
￿
1i (see Fig.15).
This OR corresponds to an initial lattice mismatch between
hAlf100gi and hc-Al
2
O
3
i as large as +18% (at T = 300 K)
in one direction parallel to the hAlð100Þijhc-Al
2
O
3
ð111Þi in-
terface plane and a much lower initial lattice mismatch of
about +2.0% (as for the overgrowth on hAlf111gi;see Sec-
tions 5.1.and Fig.13) in the perpendicular direction.Accord-
ing to the LEED and HRTEM analysis,the large anisotropic
tensile growth strain in the hc(-like)-Al
2
O
3
i overgrowth on
hAlf100gi has predominantly been relaxed by the formation
of defects at the incoherent hAlð100Þijhc-Al
2
O
3
ð111Þi inter-
face,as well as by slight,in-plane rotations (of about ±48) of
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Fig.13.HRTEM micrograph and corresponding LEED pattern (at
53 eV) of the crystalline hc(-like)-Al
2
O
3
i overgrowth on hAlf111gi
after oxidation for t = 6000 s at T = 373 K and p
O
2
= 1∙ 10
– 4
Pa (and
subsequent in-situ deposition of a MBE-grown Al seal after in-situ
LEEDanalysis).The direction of the primary electron beamwas 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 en-
ergy 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-Al
2
O
3
i [56].
Fig.14.Calculated critical thickness (h
crit
fM
x
O
y
g
) up to which an amor-
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 or-
dinate indicates the corresponding critical thickness in oxide mono-
layers (MLs) as obtained by taking 1 oxide ML ffi0.22 nm (see also
Refs.[2,3,41,42,104]).
two types of hc(-like)-Al
2
O
3
i 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)-Al
2
O
3
i overgrowth and its parent
hAlf100gi substrate can be explained considering the
surface-energy and interface-energy contributions for the
hc-Al
2
O
3
i 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 be-
tween overgrowth and hAlf100gi,and for the originally ex-
pected 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 with-
in 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,
c
mismatch
hAlf100gijhc
-
Al
2
O
3
f111gi
,due to the sumof the residual growth
strain and misfit dislocations in the hc-Al
2
O
3
i overgrowth
(Section 4.3.) is considerably larger (i.e.more positive) for
the high-mismatch OR.However,the corresponding inter-
action energy contribution,c
interaction
hAlf100gijhc
-
Al
2
O
3
f111gi
,is much
more negative (as compared
to c
interaction
hAlf100gijhc
-
Al
2
O
3
f100gi
) due to a higher density of metal-
oxygen bonds across the hAlð100Þijhc-Al
2
O
3
ð111Þi inter-
face (than across the hAlð100Þijhc-Al
2
O
3
ð100Þi interface).
Since the (negative) interaction energy,c
interaction
hAlf100gijhc
-
Al
2
O
3
f111gi
,
is the dominant energy contribution to the interface energy
(Section 4.3.),the larger (positive) mismatch contribution,
c
mismatch
hAlf100gijhc
-
Al
2
O
3
f111gi
(as compared to c
mismatch
hAlf100gijhc
-
Al
2
O
3
f100gi
),
is overcompensated by its more negative interaction energy
contribution,c
interaction
hAlf100gijhc
-
Al
2
O
3
f111gi
.In addition,the energy
of the hc-Al
2
O
3
(111)i surface is much lower than that
of the less-densely-packed hc-Al
2
O
3
(100)i surface (i.e.
c
S;0
hc
-
Al
2
O
3
ð111Þi
%0.9 J m
– 2
,whereas c
S;0
hc
-
Al
2
O
3
ð100Þi
ffi 1.9 J m
– 2
)
[124].Thus the observed high-mismatch OR for the initial
hc-Al
2
O
3
i overgrowth on hAlf100gi is thermodynamically
preferred (instead of the low-mismatch OR),because of
the lower sumof the surface and interface energy contribu-
tions,in spite of the higher energy contributions due to re-
sidual strain and misfit dislocations in the corresponding
high-mismatch hc-Al
2
O
3
i overgrowth.This implies that
the generally adopted assumption,that the OR correspond-
ing with the lowest possible lattice mismatch (i.e.the “best
fit” OR) is energetically preferred,needs not hold for ultra-
thin 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 di-
rect contact with a metal,such as Al [125],Au [126],Ag
[127],Ni [128],Cu [129],and Pd [130] (Fig.17).This phe-
nomenon,which is now commonly referred to as metal-in-
L.P.H.Jeurgens et al.:Thermodynamics of reactions and phase transformations at interfaces and surfaces
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Fig.15.HRTEM micrograph and corresponding LEED pattern (at
53 eV) of the high-mismatch crystalline hc(-like)-Al
2
O
3
i overgrowth
on hAlf100gi after oxidation for t = 6000 s at T = 550 K and
p
O
2
= 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-fil-
tered region of the original micrograph as obtained after inverse Four-
ier 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 sym-
metry) 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 en-
ergy,and (c) chemical interaction energy contributions to (d) the resul-
tant hAlijhc-Al
2
O
3
i interface energy,c
hAlijhAl
2
O
3
i
,as function of the oxide
film thickness,for hc-Al
2
O
3
i overgrowth on the bare hAlf100gi sub-
strate 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-Al
2
O
3
f100gi or the high-
mismatch (i.e.,hAlf100gijhc-Al
2
O
3
f111gi OR between the oxide
overgrowth and the hAlf100gi substrate.The corresponding energies
for the overgrowth of low-mismatch (i.e.,hAlf111gijhc-Al
2
O
3
f111gi
on the hAlf111gi substrate are also shown for comparison [42].
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 crystal-
line-Si-based (further designated as c-Si or hSii) thin-film
devices such as flat-panel displays and solar cells on low-
cost/flexible but usually heat-sensitive substrates,MIC has
been extensively investigated in various metal/amorphous-
semiconductor systems [17,132].
The strong covalent bonding in bulk amorphous semi-
conductors accounts primarily for their high crystalliza-
tion 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 mono-
layers (ML) thick [133] and is generally believed to
provide the agent for initiation of crystallization of amor-
phous semiconductors at low temperatures.By consider-
ing quantitatively the interface energetics related to these
2-ML interfacial “free” atoms (i.e.the competition be-
tween the change of the “bulk” energies and the change
of the corresponding surface and interface energies) upon
initiation of MIC,the different MIC temperatures/behav-
iours in various immiscible metal/amorphous-semicon-
ductor systems can be understood and predicted on a uni-
fied 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 ther-
modynamically on whether the total interface energy can be
reduced by replacing the GB with two interphase bound-
aries,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 Sec-
tions 4.1.,4.2.1.,and 4.3.1.,the energetics for possible GB
wetting processes in many crystalline-metal/amorphous-
semiconductor (hMijfSg) systems have been evaluated
quantitatively.As shown in Fig.18,the energy of a high-
angle GB in the crystalline metal,hMi,(i.e.the value of
e.g.c
GB
hAli
,c
GB
hAgi
or c
GB
hAui
,as evaluated on the basis of
Eq.(39) in Section 4.3.1.) can indeed be substantially larger
than the sumof 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.
c
hAlijfSig
,c
hAlijfGeg
,c
hAgijfSig
or c
hAuijfSig
,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 inter-
face with the metal and/or at the wetted metal GBs (Sec-
tion 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 inter-
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Fig.17.The reduction in the crystallization temperature,T
crys
,of a-Si
induced by contact with various metals [132].
(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
GBs and hAgi GBs by a-Si.Positive driving forces (c
GB
hMi
2  c
hMijfSg
)
are evident for the occurrence of GB wetting in these systems [17].
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 inter-
face(s) or within the GB of a contacting crystalline metal
layer can be thermodynamically stable up to a certain criti-
cal 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 con-
straint,that the critical thickness should be smaller than
2 MLs in order that crystallization can initiate at a crystal-
line-metal/amorphous-semiconductor interface at relatively
low temperatures,because the layer of “free” semiconduc-
tor atoms at the interface has a maximal thickness of only
2 ML (see the introductory part of Section 6).
The critical thickness,h
crit
hMijfSg
,for the nucleation of a
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 cor-
responding decrease of the “bulk” Gibbs energy (as given
by the bulk crystallization energy,DG
crystallization
hSi!fSg
,in J m
–3
):
h
crit
hMijfSg
ðTÞ ¼
c
hMijhSi
ðTÞ þc
hSijfSg
ðTÞ c
hMijfSg
ðTÞ
DG
crystallization
hSi!fSg
ðTÞ
ð48Þ
It is important to recognize that h
crit
hMijfSg
ðTÞ should be smal-
ler 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 con-
tacting 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
crit
hMijfSgjhMi
ðTÞ ¼
2  c
hMijhSi
ðTÞ c
hMijfSg
ðTÞ
h i
DG
crystallization
hSi!fSg
ðTÞ
ð49Þ
The wetting fSg film at the metal GBs is sandwiched be-
tween 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,
h
crit
hMijfSgjhMi
ðTÞ must be smaller than (or maximally equal
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 func-
tion of T for various metal/amorphous-semiconductor sys-
tems 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
c
hMijfSg
and c
hMijhSi
) were evaluated as a function of T using
Eqs.(30) and (40) in Sections 4.2.1.and 4.3.1.,respec-
tively.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 func-
tion of temperature in Fig.19a.
It follows that the calculated critical thickness for crystal-
lization of a-Si at the hAlijfSig interface is larger than
2 ML up to 4008C and beyond.This implies that initia-
tion of Al-induced crystallization at the hAlijfSig interface
is thermodynamically impossible.At the hAli GBs for
T > 1408C,on the other hand,the critical thickness for crys-
tallization of the wetting a-Si film is below 4 ML.Hence,
for the hAli–fSig layer system,the only site for c-Si to nucle-
ate at low temperatures is the Al GB with a predicted tem-
perature for the onset of crystallization > 1408C.Indeed,ex-
perimental studies of the MIC process in hAli–fSig layer
systems have indicated a minimal temperature for the onset
of crystallization of 1508C [17,135].It has also been con-
firmed 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 oc-
cur both at the hAlijfGeg interface and at the Al GBs (for
L.P.H.Jeurgens et al.:Thermodynamics of reactions and phase transformations at interfaces and surfaces
1302 Int.J.Mat.Res.(formerly Z.Metallkd.) 100 (2009) 10
F
Feature
(a)
(b)
Fig.19.(a) Calculated critical thicknesses for nucleation of c-Si and c-
Ge 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 *508C.(b) Calculated criti-
cal 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) *4008C.For the hAui–fSig system,it follows
that c-Si cannot nucleate directly at hAui GBs and at the hAuijfSig inter-
face.Instead,MICin hAui–fSig layer systems is mediated by the forma-
tion of metastable hAu
3
Sii phase nucleated at hAui GBs [17].
T > 508C).This prediction is fully consistent with corre-
sponding 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 > 4008C.This
predicted MIC behavior also agrees very well with experi-
mental 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,ex-
periments have shown that the MIC process in the
hAui–fSig system is mediated by the formation of a meta-
stable hAu
3
Sii silicide phase at a very low temperature of
*1008C [126].Therefore,thermodynamic calculations,
which account for the nucleation of hAu
3
Sii at the
hAuijfSig interface and at the hAui GBs,have also been
carried out (Fig.19b).Indeed,the theoretical critical thick-
ness for the formation of hAu
3
Sii at fSig-wetted hAui GBs
is equal or lower than 4 ML for T > 808C,which is well
compatible with the observed formation of hAu
3
Sii at hAui
GBs in the hAui–fSig systemat *1008C [126].
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 crystal-
lize there.The driving force for this secondary wetting pro-
cess is given by:
Dc
fSg!hMijhSi
¼ c
hMijhSi
ðc
hMijfSg
þc
hSijfSg
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
growlaterally,i.e.perpendicular to the hMijhSi bound-
aries,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
crit
hSi graingrowth
¼
c
hMijhSi
ðTÞ  c
hMijfSg
ðTÞ þc
hSijfSg
ðTÞ
h i
DG
crystallization
hSi!fSg
ðTÞ
ð51aÞ
Ad (ii):the critical thickness for the formation of new hSi
grains at the hMijhSi boundaries,is given by Eq.(51b):
(see bottomof page)
The calculated critical thicknesses for the hAli–fSig sys-
temaccording to Eqs.(51a) and (51b) are plotted as a func-
tion 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 thick-
ness of the “free” fSig atoms adjacent to hAli metal is only
about 2 ML,it follows that the formation of newc-Si nuclei
at the hAlijhSii boundaries is impossible at low tempera-
tures.Instead,the critical thickness for continued c-Si grain
growth according to Eq.(51a) is only *1.5 ML (at
T > 1508C).Hence,continued lateral growth of the c-Si
grains initially formed at the original hAli GBs is possible,
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h
crit
hSi newnucleation
¼
c
hMijhSi
ðTÞ þc
hSijhSi
ðTÞ  c
hMijfSg
ðTÞ þc
hSijfSg
ðTÞ
h i
DG
crystallization
hSi!fSg
ðTÞ
ð51bÞ
(a)
(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 wet-
ting 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].
which is in accordance with the in-situ and ex-situ TEM
analyses.Since the driving force,Dc
fSg!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 (lat-
eral grain growth) of Si within the original Al GBs even-
tually 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].
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 illus-
tration for the hAlijfSig layer system in Fig.21a.For such
small values of the Al overlayer thickness,h
hAli
,and the
wetting fSig film,h
fSig
,it follows that upon initiation of
crystallization of the wetting fSig film at the hAli GBs
(taken as running perpendicular to the film surface:colum-
nar grain structure),not only the interface energy change,
2  h
hAli
 ðc
hAlijhSii
 c
hAlijfSig
Þ,associated with the replace-
ment of the two original hAlijfSig interfaces by two
hAlijhSii interfaces,but also the surface and interface en-
ergy changes,h
fSig
 ðc
S
hSii
c
S
fSig
Þ and h
fSig
 c
hSiijfSig
,as-
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 initia-
tion of crystallization of a-Si at the Al GBs as function of both
T and h
hAli
is given by [4] (Eq.(52)):(see bottomof page)
The critical thickness,h
crit
hAlijfSigjhAli
,as function of both
h
hAli
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
value of h
crit
hAlijfSigjhAli
must be smaller than (or maximally be
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
the solid line pertaining to h
crit
hAlijfSigjhAli
= 4 ML in Fig.21b.
It follows that the crystallization temperature is around
150–2008C for hAli overlayer thicknesses h
hAli
> 20 nm.
For h
hAli
< 20 nm,the crystallization temperature increases
strongly with decreasing h
hAli
.
These theoretical predictions have been experimentally
confirmed by monitoring the crystallization behavior of
a-Si as a function of the hAli overlayer thickness under ul-
tra-high vacuum conditions by real-time in-situ spectro-
scopic ellipsometry [4].The corresponding crystallization
temperature of a-Si decreases from about 7008C to 1808C
for an increase of the thickness h
hAli
of the covering Al film
fromh
hAli
< 1 nmto h
hAli
= 20 nm(Fig.21b).
7.Conclusion
The contributions of surface and interface energies to the
total Gibbs energy can predominate the energetics of low-
dimensional systems.On the basis of this recognition the
formation of many experimentally observed,nano-size-
related microstructures,which are in flagrant contrast
L.P.H.Jeurgens et al.:Thermodynamics of reactions and phase transformations at interfaces and surfaces
1304 Int.J.Mat.Res.(formerly Z.Metallkd.) 100 (2009) 10
F
Feature
h
crit
hAlijfSigjhAli
ðT;h
Al
Þ ¼
2  c
hAlijhSii
ðTÞ c
hAlijfSig
ðTÞ
h i
DG
crystallization
hSii!fSig
ðTÞ 
c
S
hSii
ðTÞ c
S
fSig
ðTÞ
h i
þ
c
hSiijfSig
ðTÞ
h
hAli
ð52Þ
(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 over-
layer.(b) Calculated critical thickness for initiation of crystallization of
a-Si at hAli GBs as functions of both the hAli overlayer thickness,h
hAli
,
and the temperature,T.The line pertaining to h
critical
hAlijfSigjhAli
= 4 ML gives
a theoretical prediction of the crystallization temperature of a-Si as func-
tion of the hAli overlayer thickness.The corresponding experimental
confirmation of the dependence of the crystallization temperature of a-
Si on h
hAli
has also been indicated [4,132].
with expectations derived frombulk 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 crystal-
line 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 pro-
vides quantitative estimates for the thickness of these amor-
phous product layers beyond which crystallization of a
stable crystalline phase should occur.All these predictions
are in (quantitative) agreement with experimental observa-
tions.
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 crys-
tallization 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 crystalliza-
tion.We thank Dr.G.Richter for HRTEM analysis of the ultra-thin
oxide overgrowths and Dr.P.A.van Aken for provision of TEMfacil-
ities.
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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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L.P.H.Jeurgens et al.:Thermodynamics of reactions and phase transformations at interfaces and surfaces
Int.J.Mat.Res.(formerly Z.Metallkd.) 100 (2009) 10 1307
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