# Adhesion and material transfer between contacting Al and TiN surfaces from
first principles

G. Feldbauer<EMAIL_ADDRESS>Institute of Applied Physics, Vienna
University of Technology, Gußhausstraße 25-25a, 1040 Vienna, Austria  Austrian
Center of Competence for Tribology, AC2T research GmbH, Viktor-Kaplan-Straße
2, 2700 Wiener Neustadt, Austria  M. Wolloch  Institute of Applied Physics,
Vienna University of Technology, Gußhausstraße 25-25a, 1040 Vienna, Austria
Austrian Center of Competence for Tribology, AC2T research GmbH, Viktor-
Kaplan-Straße 2, 2700 Wiener Neustadt, Austria  P. O. Bedolla  Institute of
Applied Physics, Vienna University of Technology, Gußhausstraße 25-25a, 1040
Vienna, Austria  Austrian Center of Competence for Tribology, AC2T research
GmbH, Viktor-Kaplan-Straße 2, 2700 Wiener Neustadt, Austria  P. Mohn
Institute of Applied Physics, Vienna University of Technology, Gußhausstraße
25-25a, 1040 Vienna, Austria  J. Redinger  Institute of Applied Physics,
Vienna University of Technology, Gußhausstraße 25-25a, 1040 Vienna, Austria
A. Vernes  Institute of Applied Physics, Vienna University of Technology,
Gußhausstraße 25-25a, 1040 Vienna, Austria  Austrian Center of Competence for
Tribology, AC2T research GmbH, Viktor-Kaplan-Straße 2, 2700 Wiener Neustadt,
Austria

###### Abstract

A series of density functional theory (DFT) simulations was performed to
investigate the approach, contact, and subsequent separation of two atomically
flat surfaces consisting of different materials. Aluminum (Al) and titanium
nitride (TiN) slabs were chosen as a model system representing a metal-ceramic
interface and the interaction between soft and hard materials. The approach
and separation were simulated by moving one slab in discrete steps normal to
the surfaces allowing for electronic and atomic relaxations after each step.
Various configurations were analyzed by considering (001), (011), and (111)
surfaces as well as several lateral arrangements of these surfaces at the
interface. Several tests were conducted on the computational setup, for
example, by changing the system size or using different approximations for the
exchange correlation functional. The performed simulations revealed the
influences of these aspects on adhesion, equilibrium distance, and material
transfer. These interfacial properties depend sensitively on the chosen
configuration due to distinct bond situations. Material transfer, in
particular, was observed if the absolute value of the adhesion energy for a
given configuration is larger than the energy cost to remove surface layers.
This result was found to be independent of the employed exchange correlation
functional. Furthermore, it was shown that a simple comparison of the surface
energies of the slabs is not sufficient to predict the occurrence of material
transfer.

###### pacs:

31.15.E-, 81.07.Lk, 62.20.Qp, 71.15.Mb

##  I Introduction

Contacts of surfaces at the atomic length scale are crucial in many modern
applications, from experimental techniques such as nanoindentation Landman
_et al._ (1990); Oliver and Pharr (2004); Fischer-Cripps (2004) or
atomic/friction force microscopy (AFM/FFM) Binnig  _et al._ (1986); Meyer  _et
al._ (2004); Bennewitz (2005) to nanotechnologies applied, for example, in
nano-/microelectromechanical-systems (NEMS/MEMS) Komvopoulos (1996); Spearing
(2000); Maboudian and Carraro (2004); Kim  _et al._ (2007); Bhushan (2008).
The reliability, performance, and lifetime of such systems, for example,
depend sensitively on the interactions between contacting materials.
Furthermore, detailed insights into such contacts are of fundamental interest
for better comprehension of tribological processes, such as nanoscale wear
Bhushan  _et al._ (1995); Gnecco  _et al._ (2002); Bhushan (2005); Gnecco
(2007); Gotsmann and Lantz (2008); Bhaskaran  _et al._ (2010); Jacobs and
Carpick (2013); Mishina and Hase (2013), for which there is still a lack of
understanding due to its highly complex nature Kim  _et al._ (2012).

Metal-ceramic interfaces Howe (1993a) are of fundamental and technological
interest because they exhibit advantages of both types of materials such as
valuable mechanical properties, high thermal stability, and degradation
resistance Johansson (1995). Hence, such interfaces are important in numerous
applications such as communication devices and nanoelectronics Rühle  _et al._
(1992). In this paper the interface between the metal Al and the transition-
metal nitride TiN is investigated. This interface consists of a soft material
and a hard material, which simplifies wear processes because the softer
material is primarily affected.

Since the 1980s classical molecular dynamics (MD) simulations have commonly
been applied to nanotribological problems (see, e.g., Refs. Thompson and
Robbins, 1989; Landman  _et al._ , 1989; Robbins and Müser, 2000; Mulliah  _et
al._ , 2004; Kenny  _et al._ , 2005; Schall  _et al._ , 2007; Szlufarska  _et
al._ , 2008; Vernes  _et al._ , 2012; Eder  _et al._ , 2013, 2014a, 2014b) and
still constitute a standard tool in numerical atomistic simulations.
Nevertheless, during the last decade density functional theory (DFT)
calculations have been increasingly used in this field (see, e.g., Refs. Zhong
and Tománek, 1990; Dag and Ciraci, 2004; Ciraci  _et al._ , 2007; Zilibotti
_et al._ , 2009; Garvey  _et al._ , 2011; Zilibotti and Righi, 2011;
Cahangirov  _et al._ , 2012; Kwon  _et al._ , 2012; Wang  _et al._ , 2012a, b;
Garvey  _et al._ , 2013; Wolloch  _et al._ , 2014) and should be seen as an
extension to the more common computational tools in tribology. DFT allows for
parameter-free calculations and an accurate description of quantum-mechanical
systems and does not depend on empirical potentials. However, due to
computational challenges DFT calculations are currently limited to rather
small systems of typically a few hundred atoms. Since DFT has proven to yield
reliable results for this class of systems Finnis (1996); Lundqvist  _et al._
(2001); Sinnott and Dickey (2003), it is also employed in this study to
analyze the electronic and atomic structure of the Al/TiN interfaces and to
determine properties such as adhesion energies. Results obtained with DFT,
such as potential-energy curves, can be used as an input for, e.g., large-
scale classical MD simulations Ercolessi and Adams (1994); Jaramillo-Botero
_et al._ (2014). Furthermore, the combination of approaches such as DFT and MD
as well as the continuously increasing available computer power and advances
in software tools promise the possibility to investigate even larger and more
realistic systems in the near future.

Al/TiN and similar interfaces have already been investigated by various
researchers with experimental Avinun  _et al._ (1998); Chun  _et al._ (2001)
and theoretical Liu  _et al._ (2003, 2004, 2005); Song and Srolovitz (2006);
Zhang  _et al._ (2007); Zhang and Wang (2007); Song and Srolovitz (2008);
Yadav  _et al._ (2012, 2014) methods. Here, however, the emphasis lies on a
realistic way to simulate the separation of the interfaces as well as on a
comprehensive discussion of interfaces between Al and TiN low-index surfaces.
To assess this problem, the effects of various configurations at the interface
as well as approach and subsequent separation of Al and TiN slabs are
analyzed. Various tests on, e.g., the effect of adjusted lattice parameters,
the simulation cell size, and various approximations for the exchange
correlation functional in DFT are carried out.

##  II Computational Details

###  II.1 Density Functional Theory Calculations

To study the interfacial properties of Al and TiN slabs upon approach and
subsequent separation, we performed first-principles calculations within the
framework of DFT employing the Vienna Ab initio Simulation Package (VASP)
Kresse and Hafner (1993, 1994); Kresse and Furthmüller (1996a, b). VASP
utilizes a plane-wave basis and periodic boundary conditions. Projector
augmented-wave (PAW) pseudopotentials Blöchl (1994); Kresse and Joubert (1999)
were used to model the potential between the ionic core and the valence
electrons. Unless explicitly mentioned otherwise, the generalized gradient
approximation (GGA) in the Perdew, Burke, and Ernzerhof (PBE) parametrization
Perdew  _et al._ (1996) was applied to describe the exchange and correlation
functional. Since GGAs often underestimate binding and adhesion energies
Stampfl and Van de Walle (1999), the local-density approximation (LDA) Perdew
and Zunger (1981), which usually overestimates these quantities van de Walle
and Ceder (1999), was also employed for comparison. Additionally, the van der
Waals (vdW) density functional (DF) optB86b Klimeš  _et al._ (2010, 2011) was
used, which includes a nonlocal correlation term approximating vdW
interactions. vdW-DFs have been applied to a wide range of materials (e.g.,
see Refs. Chakarova-Käck  _et al._ , 2006; Sony  _et al._ , 2007; Carrasco
_et al._ , 2011; Mittendorfer  _et al._ , 2011; Antlanger  _et al._ , 2012;
Graziano  _et al._ , 2012; Bedolla  _et al._ , 2014a; Choi  _et al._ , 2014;
Bedolla  _et al._ , 2014b) and have proven to be of good accuracy. Although
vdW interactions should not play a major role in the investigated systems, the
calculations are included for comparison and clarification. The calculation
parameters were carefully chosen to obtain accurate total energies. An energy
cutoff of 800 eV was used for the plane-wave basis. Unless noted otherwise,
the Brillouin zone sampling was performed using a $\Gamma$-centered $15\times
15\times 1$ Monkhorst-Pack mesh Monkhorst and Pack (1976). Both settings allow
for total energies accurate to 1 meV/atom. While the tetrahedron method with
Blöchl corrections Blöchl  _et al._ (1994) was utilized for static
calculations, for relaxations a smearing of 0.11 eV using the first-order
method of Methfessel and Paxton Methfessel and Paxton (1989) was selected. In
order to relax the structures a damped molecular dynamics (MD) algorithm was
employed, allowing for atomic movements until an energy convergence criterion
of $10^{-5}$ eV was fulfilled. This damped MD scheme was chosen instead of the
widely used quasi-Newton or conjugate-gradient algorithms, because these
caused convergence problems as well as the tendency to remain stuck in local
minima. Each converged relaxation was followed up by a static calculation to
obtain more accurate total energies. For electronic self-consistency cycles a
convergence criterion of $10^{-6}$ eV was used. All simulations were performed
at 0 K.

###  II.2 Simulation Model

To model our systems we built simulation cells from a fcc Al slab at the
bottom and a rock salt TiN slab above (see Fig. 1). Such cells were
constructed for the low-index surface orientations (001), (011), and (111) of
both slabs. Only configurations with slabs of the same surface orientations at
the interface and without any relative rotations were considered. The two
slabs were separated by a gap which is given by the vertical distance between
the top Al and bottom TiN layers and will be referred to as the “interface
distance”. The vertical distance between the bottom Al and top TiN layers,
which is the sum of the interface distance and the heights of the two slabs,
is called “slab height”. In the case of (111) slabs this height is measured up
to the top Ti and N layer for Ti and N termination, respectively. Unless
otherwise stated 1$\times$1 surface cells were used, which represent an
infinitely extended surface due to the periodic boundary conditions. The Al
slab consisted of at least seven layers, and the TiN slab consisted of a
minimum of six Ti and six N layers. These thicknesses were found to be
sufficient to converge the surface energies and to mimic bulklike features in
the center of the respective slab. These system dimensions are in good
agreement with other published work Marlo and Milman (2000); Liu  _et al._
(2004); Zhang  _et al._ (2007); Yadav  _et al._ (2014).

Figure 1: Side view of a (111) Al/TiN interface (TiN: Ti terminated). The
simulation interface cell is indicated by the solid black lines. During
relaxations the orange Al, cyan N and purple Ti atoms were kept rigid, while
the red Al, green N and blue Ti ones were allowed to relax.

The (111) TiN slab can be terminated with either Ti or N atoms. To investigate
the stability of these terminations a thermodynamic analysis Wang  _et al._
(1998); Reuter and Scheffler (2001) was performed to calculate the surface
Gibbs free energy for the off-stoichiometric slabs Lee  _et al._ (2011). The
surface Gibbs free energy $\Omega$ for surface termination $i$ without
vibrational contributions is given by

$\Omega^{i}=\frac{1}{2}\left(E^{i}_{slab}-N^{i}_{Ti}E^{bulk}_{TiN}\right)-\Gamma^{i}_{Ti,N}E_{N}-\Gamma^{i}_{Ti,N}\Delta\mu_{N},$
(1)

where $E^{i}_{slab}$ is the total energy of the slab with termination $i$,
$N^{i}_{Ti}$ isthe number of Ti atoms in the slab, $E^{bulk}_{TiN}$ is the
total energy of bulk TiN, and $E_{N}$ is the total energy of a nitrogen atom.
The two latter terms in Eq. (1) are necessary to calculate the surface energy
of off-stoichiometric slabs. The number of off-stoichiometric atoms
$\Gamma^{i}_{Ti,N}$ is defined as

$\Gamma^{i}_{Ti,N}=\frac{1}{2}\left(N^{i}_{N}-N^{i}_{Ti}\frac{N^{bulk}_{N}}{N^{bulk}_{Ti}}\right),$
(2)

where $N^{i}_{j}$ and $N^{bulk}_{j}$ are the number of atoms of type j in the
slab and in bulk, respectively. For rock-salt bulk TiN the fraction
$N^{bulk}_{N}/N^{bulk}_{Ti}$ in Eq. (2) is equal to 1. $\Delta\mu_{N}$ is the
deviation of the nitrogen chemical potential $\mu_{N}$ from the molecular
reference $\frac{1}{2}E_{N_{2}}$,

$\Delta\mu_{N}=\mu_{N}-\frac{1}{2}E_{N_{2}}.$ (3)

In Figure 2 the calculated surface Gibbs free energy is plotted for the N- and
Ti-terminated TiN (111) slabs in the stability range of nitrogen in TiN
obtained from the heat of formation of bulk TiN Chase (1998) at 0 K, $\Delta
H^{0}_{f,(TiN)}=-3.461$ eV, and the chemical potential of gas phase nitrogen,
i.e., $\Delta H^{0}_{f}\leq\Delta\mu_{N}\leq 0$. Fig. 2 shows that the
favorable termination of a (111) TiN slab depends on the chemical potential of
nitrogen, in agreement with the Refs. Liu  _et al._ , 2004; Wang  _et al._ ,
2010. Since both cases are found in reasonable nitrogen concentration ranges,
both terminations are investigated.

Dipole corrections Neugebauer and Scheffler (1992) perpendicular to the
interface (z direction) were tested for the systems but were found to be
negligible. Atop the TiN slab a vacuum spacing of at least 10 Å was included
to decouple periodically repeated cells in the z direction. The lattice
parameters of the single slabs, 4.04 Å and 4.254 Å for Al and TiN,
respectively, were obtained from bulk calculations. These values are in very
good agreement with the experimental lattice constants of 4.05 Å and 4.265 Å
for Al and TiN, respectively Wyckoff (1963). The relative error between
calculated and experimental values is below 0.5%. For the simulation cells
combining Al and TiN slabs, unless otherwise stated, an intermediate lattice
parameter of 4.144 Å was used for the lateral xy lattice vectors to equalize
the relative error of about 2.6% for both materials. For the z direction the
material-specific values were kept assuming a pseudomorphic interface. In
reality such a combination of stretching and compression of thick slabs does
not usually occur, but dislocations at the interface or an incommensurate
contact are possible. Thus, some of the atoms on both sides of the interface
would not be aligned perfectly, but rather sample slightly different local
environments. For computational reasons, here these different local
arrangements are assessed by considering various orientations at the interface
as limiting cases.

Figure 2: Surface phase diagram for TiN. The surface Gibbs free energy
$\Omega$ [see Eq. (1)] referenced to a 1$\times$1 surface cell of the (111)
orientation is plotted vs the deviation $\Delta\mu_{N}$ of the nitrogen
chemical potential from its molecular reference [see Eq. (3)] for N- and Ti-
terminated (111) TiN slabs (solid lines) as well as for (001) and (011)
orientations (dashed lines).

The approach of the two slabs was simulated by moving the upper slab in
discrete steps along the negative z direction and allowing for electronic and
atomic relaxations after each step. Alternatively, moving the bottom slab
toward the upper slab or both toward each other would not affect the results.
For the atomic relaxations the top TiN (three Ti and three N) and the bottom
three Al layers were kept fixed at bulklike distances, whereas the
intermediate “free” ones were allowed to fully relax. This is depicted in Fig.
1 for the Ti-terminated (111) surface orientation. For the approaching
movement a step size of 0.2 Å was used for all configurations. Before the
slabs were brought into contact, the free layers were allowed to relax in
order to simulate surfaces in their equilibrium for the chosen lattice
parameters. The separation of the slabs was initiated from the equilibrium,
i.e., the structure with the lowest energy determined during the approach. To
simulate a realistic separation of the slabs only the topmost, rigid TiN
layers were moved in discrete steps in the positive z direction, again
allowing for electronic and atomic relaxations after each step. The choice of
the step size is crucial for the separation process. Separation velocities
allowing for an adiabatic behavior of the system were assumed, meaning that
the system continuously fully adjusts during the separation at each step.
However, this assumption should also be valid for velocities up to several
hundred meters per second as long as these are still considerably lower than
the material-specific speed of sound, which is above 6000 m/s for Al and TiN
Kundu (2012). It is evident that the step size has to be small enough to mimic
the adiabatic relaxation, but on the other hand, a smaller step size leads to
increased computational costs. For the investigated systems a step size of 0.1
Å was found to be a practical trade-off because calculations showed this value
to be necessary to converge the final results of the simulated separation
processes. Smaller step sizes down to 0.01 Å were also considered for approach
and separation but did not yield qualitatively different results. Clearly,
quantities such as the slab height corresponding to the initial material
transfer can be determined more accurately.

In order to study the effects of different alignments of the slabs at the
interface, the upper slab was also laterally placed on various sites with
respect to the surface of the lower slab. The definitions of the
configurations are depicted in Fig. 3 by marking the high-symmetry points on
the low index TiN surfaces where the next Al atom can be placed. In this
context the interaction energy $E_{I}(z)$ is an important quantity, which is
defined as the difference of the total energy of the interacting slabs
$E_{(Al/TiN)}(z)$ at slab height $z$ and the reference energies of the
independent slabs, $E_{(Al)}$ and $E_{(TiN)}$,

$E_{I}(z)=E_{(Al/TiN)}(z)-E_{(Al)}-E_{(TiN)}.$ (4)

Figure 3: Top view of (001), (011), and Ti-terminated (111) TiN surfaces. For
each orientation the 1$\times$1 surface cell is presented. Filled circles
indicate atoms in the top surface layer (Ti and N are given by large blue and
small green circles, respectively), while empty circles label atoms below the
top surface layer. To obtain a N-terminated (111) TiN surface the Ti and N
atoms of the Ti-terminated surface have to be exchanged. High-symmetry points
are highlighted. For the (011) TiN surface the “Ti plane” and “N plane” are
marked by dashed lines.

##  III Results and Discussion

###  III.1 Removal of Layers from an Al Slab

As a first step the energy cost for removing layers from an Al slab was
examined for all three low-index surface orientations. The removal of the
layers was simulated by placing the layers at a large distance from the slab,
which does not allow for interactions between the slab and layers. The TiN
slab is not investigated here because the Al slab is assumed to be mainly
affected by deformations or material transfer within an Al/TiN interface
because TiN forms a much more rigid lattice. The energetical results for the
removal of the top Al layer are given in Table 1. These removal energies are
calculated for simulation cells using the bulk lateral lattice parameters as
well as the modified ones used for the Al/TiN simulation cell. For the
modified lattice parameters the removal energies are typically overestimated
by about 5%–10%, meaning that it is actually easier to remove layers from the
equilibrium structure. The removal energy for the modified Al slab is
increased because the lateral stretching causes a vertical compression of the
slab if relaxations are allowed. This compression occurs to minimize the
volume change and locally strengthens the bonding of the surface layers. This
effect is strongest for the top surface layer, which moves about 0.24 Å
towards the rigid part and becomes weaker for the subsurface layers; for
example, the fourth layer is only shifted by about 0.08 Å.

The influence of compressive and tensile stress on the removal energies of the
top Al layer is illustrated in Fig. 4 for the three low-index surface
orientations. The data points for the (001) and (111) surfaces follow a
similar trend, whereas the behavior of the (011) surface clearly deviates.
This difference occurs probably due to the openness of the (011) surface and
the significant impact of relaxations. The influence of stress, found for all
surfaces, supports the notion of stress-assisted wear Gotsmann and Lantz
(2008); Jacobs and Carpick (2013), which states the possibility of a reduction
of the activation barriers for the detachment of atoms from a structure due to
stress. Furthermore, different approximations for the exchange correlation
functional were tested. As expected it was found that LDA and the vdW-DF
optB86b yield larger removal energies by about 15%–20%, where LDA typically
gives values larger by a few percent than the vdW-DF (see Table 1).

Table 1:  Energy costs to remove the top layer from an Al slab for the (001), (011), and (111) surface orientations using PBE, LDA, and vdW-DF optB86b. The removal energies are given in eV per 1$\times$1 surface cell. $a_{Al}$ is the Al bulk lattice parameter, whereas $a_{Al/TiN}$ corresponds to the modified Al/TiN interface. | (001) | (011) | (111)
---|---|---|---
PBE ($a_{Al}$) | 1.08 | 1.78 | 0.78
PBE ($a_{Al/TiN}$) | 1.16 | 1.79 | 0.87
LDA ($a_{Al/TiN}$) | 1.33 | 2.02 | 1.02
vdW ($a_{Al/TiN}$) | 1.27 | 1.89 | 1.00
Figure 4: PBE energy costs to remove the top Al layer for the (001), (011),
and (111) surface orientations. The lateral effects of stretching and
compression of the 1$\times$1 surface cell on the removal energies are shown.
The Al bulk lattice constant is used as a reference value at 0%. The vertical
line indicates the intermediate Al/TiN interface lattice parameter, while the
other solid lines are given to guide the eye.

###  III.2 Lateral Alignments at the Al/TiN Interface

Effects of various lateral alignments of the slabs at the interface (see Fig.
3) were investigated for the different surface orientations. These studies
revealed the strong dependence of equilibrium properties such as adhesion
energies and the equilibrium distances on the chosen configuration. The
calculated interaction energies [Eq. (4)] of relaxed interfaces are shown in
Figs. 5(a)–(d) for slab heights around the energy minima, which are equivalent
to the adhesion energies for each alignment. In general, the top Al atoms
prefer the proximity of N atoms over Ti. The bonding situation will be
discussed in more detail in the following paragraphs. From an energetical
point of view material transfer between the slabs should be possible only if
the energy cost to remove layers is compensated for. Thus, the energy gain due
to adhesion has to be larger than the energy cost to remove one or more
layers. This argument is sketched in Figs. 5(a)–(d) by including a horizontal
line at the negative value of the Al removal energy for each surface
orientation. It has been observed experimentally that metal-ceramic interfaces
with weak and strong interfacial adhesion break upon stress at the interface
and in bulk areas, respectively Howe (1993b); Ernst (1995). We find that the
four surfaces investigated exhibit essentially different behavior. The
adhesion energies and the equilibrium distances, i.e., the interface distances
at the minimum of each energy curve, depend strongly on the surface
orientation as well as on the alignment at the interface. In the case of the
(111) surfaces all configurations should lead to the removal of at least one
Al layer. For the (011) surfaces this is the case only for three alignments
(see Fig. 3), Al/N (top), Al/TiN (hollow), and Al/N (bridge). In contrast, for
the (001) surfaces no material transfer should occur since for all cases
studied the energy to remove one Al layer is larger than the adhesion energy.

(a) (111) Ti-terminated

(b) (111) N-terminated

(c) (011)

(d) (001)

Figure 5: Calculated PBE interaction energies of the relaxed Al/TiN interface
for the (111) Ti-terminated, (111) N-terminated, (011), and (001) surface
orientations. Various lateral alignments of the two slabs are considered (see
Fig. 3). The horizontal, dashed orange lines give the energy costs to remove
at least one layer from an Al slab of the corresponding surface orientation.

As mentioned above, in reality, surfaces with a bulk lattice mismatch are
usually not perfectly aligned at an interface. Consequently, not all atoms are
placed on the same contact site; therefore, the interfacial properties such as
the adhesion energy are an average of the actually occupied sites. The
configurations presented here, however, constitute limiting cases of perfectly
aligned systems, such that the properties of real interfaces should be found
within these boundaries.

Generally, relaxation effects have to be accounted for to obtain the correct
equilibrium values of the adhesion energy and the interface distance as well
as to predict the occurrence of material transfer. A comparison between the
relaxed and static results is given in Fig. 6 for the (111) surfaces. For
rather closed TiN surfaces, such as the (001) and Ti-terminated (111)
orientations [see Fig. 6(a)], relaxations typically cause only small changes
in the equilibrium quantities of the interface. Hence, computationally “cheap”
static calculations give good estimates, unless pronounced changes in the
structure of the Al slab occur. This is, for example, the case for the Al/Ti
(hollow) alignment of the (111) Al/TiN (Ti-terminated) interface, since the
interfacial Al atom relaxes towards the energetically more favorable fcc
contact site. In the case of the more open (011) surface, relaxations show
more pronounced effects for all alignments and should be taken into account.
Nevertheless, the energy hierarchy and the prediction of the occurrence of
material transfer are not affected for all alignments with the exception of
the Al/TiN (hollow) case. Again, the Al/TiN (hollow) interface behaves
differently because the relaxed structure of the Al slab is modified by the
approaching TiN slab. In more detail the interfacial Al layer is moved to the
Al/N (bridge) site, which is the most favorable alignment. This movement of
about 0.8 Å occurs mainly in the lateral plane. The free subinterface layers
are shifted to gradually compensate the change in the stacking between the
fixed layers at the bottom of the slab and the interfacial layer. These shifts
range approximately between 0.2 and 0.6 Å. For the cases discussed so far,
except for the hollow alignments, relaxations showed rather small effects on
the equilibrium quantities. In contrast, all alignments of the (111) Al/TiN
(N-terminated) interface are crucially affected by relaxations [see Fig.
6(b)]. The adhesion energies are strongly increased, and the energetical
hierarchy of the alignments is altered. Furthermore, while static calculations
suggest the absence of material transfer, relaxations predict its occurrence
for all tested alignments.

(a) (111) Ti-terminated

(b) (111) N-terminated

Figure 6: Calculated PBE interaction energies of the Al/TiN interface for the
(111) Ti- and N-terminated surfaces. Solid and dashed lines indicate results
of relaxed and static calculations, respectively. The slab heights on the
lower x axis are valid for static and relaxed calculations, whereas the
interface distances on the upper x axis refer only to the static calculations.
Various lateral alignments of the two slabs are considered (see Fig. 3). The
horizontal solid orange lines give the energy costs to remove at least one
layer from an Al slab.

For a better understanding of the energetically preferred configurations at
the interface, layer-projected densities of state (DOSs) and differences in
charge densities are examined. Layer-projected DOSs are displayed in Fig. 7
for the two alignments Al/N (bridge) and Al/Ti (top) as well as the isolated
slabs of the (011) surface orientation. This surface orientation has been
chosen because it exhibits a large spread in adhesion energies for different
alignments. Additionally, the occurrence of material transfer should depend on
the alignment. In Fig. 7 “interface (surface) layers” indicate the first
layers of Al, Ti, and N immediately at the interface (surface), whereas
“subinterface (subsurface) layers” mean the next layers of Al, Ti, and N
moving deeper into both materials. Further layers are not presented because
they exhibit only minor differences with respect to the subinterface layers.
The DOSs of the shown alignments display distinct features. For the Al/Ti
(top) case, where Ti is the next interfacial neighbor of the top Al atom, the
Al DOS is almost not affected by the interface. Only a small accumulation of
sp states just below the Fermi energy and a depletion of s states at the edges
of the DOS are found for the interface layers with respect to the other
layers. The N sp states are shifted closer to the Fermi energy for the
interfacial layer, and in particular, the Ti d states exhibit more occupied
states at the Fermi energy. These changes indicate a weakly covalent bonding
between the Al sp states and the Ti d states. Furthermore, the DOS is very
similar to the case of the isolated Al and TiN slabs. This also shows the weak
interaction for the Al/Ti (top) interface. On the other hand, for the Al/N
(bridge) configuration, where the uppermost Al atoms are closer to N across
the interface, the Al DOS is changed in a more pronounced way. The sp states
in the interface layers are partially shifted to lower energies, resulting in
a pronounced peak at about -8 eV and a few minor ones around -7 eV. The N p
states around -5 eV are broadened in the interfacial layer, resulting in
common peaks with Al states roughly between -6 eV and -8 eV. These effects at
the interface indicate a hybridization of Al and N sp states and explain the
stronger adhesion due to covalent interaction. The interfacial Ti states are
only slightly affected, exhibiting a few more occupied states at the Fermi
level.

(a) Al & TiN (isolated): surface layers

(b) Al & TiN (isolated): sub-surface layers

(c) Al/Ti (top): interface layers

(d) Al/Ti (top): sub-interface layers

(e) Al/N (bridge): interface layers

(f) Al/N (bridge): sub-interface layers

Figure 7: Layer-projected DOSs from PBE calculations of the isolated Al and
TiN slabs as well as of the (011) Al/TiN interface for the Al/Ti (top) and
Al/N (bridge) alignments. The Fermi energy is shifted to 0 eV.

In addition to the DOS, charge densities at the interfaces are investigated
and presented for the same alignments of the (011) surface. For a better
visualization the differences of charge densities $\rho_{diff}$ between the
Al/TiN interface and the isolated, independent Al and TiN slabs are presented
in Fig. 8. The charge-density difference $\rho_{diff}$ is defined as

$\centering\rho_{diff}=\rho_{Al/TiN}-(\rho_{Al}+\rho_{TiN}),\@add@centering$
(5)

where $\rho_{Al/TiN}$ is the charge density of the interface, while
$\rho_{Al}$ and $\rho_{TiN}$ represent the charge densities of the isolated
slabs. Both displayed alignments result in a rather continuous charge
accumulation between Al and Ti at the interface, suggesting a bonding [see
Figs. 8(a) and 8(c)]. For the Al/N (bridge) configuration an additional charge
buildup occurs between the interfacial Al and N atoms, which indicates
covalent contributions to the bonding due to the more localized and
directional character of the accumulation [see Fig. 8(d)]. These findings
support the DOS arguments made in the previous paragraph.

(a)
Al/Ti (top): Ti-plane

(b)
Al/Ti (top): N-plane

(c)
Al/N (bridge): Ti-plane

(d)
Al/N (bridge): N-plane

Figure 8: Charge-density differences $\rho_{diff}$ [see Eq. (5)] of the (011)
Al/TiN interface. $\rho_{diff}$ was obtained from PBE calculations for the
relaxed equilibrium configurations of (a) and (b) the Al/Ti (top) alignment
and (c) and (d) the Al/N (bridge) alignment. The charge-density difference of
each alignment is plotted for the Ti plane and the N plane (recall Fig. 3) for
values from -0.2 (solid blue, deficit) to 0.2 (solid red, accumulation)
electrons/Å3. Color code: Al, orange; Ti, violet; N, cyan.

###  III.3 Approach and Separation of Al and TiN Slabs

The energetical argument on material transfer presented above can be tested by
“slowly”, i.e., using small discrete steps, approaching and subsequently
separating the slabs. The energetical results of such loops are depicted in
Figs. 9(a)–(d) for different configurations; the respective energies are
presented in Table 2. The green curves with their data points indicated by
pluses in Figs. 9(a)–(d) give static potential-energy curves, where all atoms
were kept rigid for each selected interface distance. For large interface
distances this curve shows the limiting case of separated, independent slabs.
For ever-shorter distances the effect of relaxation becomes important, and the
actual energies deviate from the green curves. The blue curves with their data
points marked by crosses show the interaction energies of the approaching
slabs including atomic relaxations after each discrete step. For some of these
cases we find rather large jumps which are either due to material transfer
between the slabs, namely, from Al to TiN, or due to the Al slab expanding
into the space between the slabs. Finally, the red curves with their data
points displayed by circles indicate the interaction energies of the
subsequent separation of the slabs, again including atomic relaxations after
each step. These curves are also not completely smooth but display some kinks
or smaller jumps mainly due to the breaking apart of the Al/TiN slab into two
separated ones. When material transfer takes place, these curves, of course,
do not approach the green ones, even at large slab separations.

(a) (111) Ti-terminated

(b) (111) N-terminated

(c) (011)

(d) (001)

Figure 9: Calculated PBE interaction energies [see Eq. 4] for the approach and subsequent separation of Al and TiN slabs for (111) Ti-terminated, (111) N-terminated, (011), and (001) surface orientations. The alignments follow the definitions in Fig. 3. Table 2:  Equilibrium interface distances, adhesion energies, energy costs to remove layers from the Al slab, and number of transferred Al layers for various interface configurations. For the (111) orientation Al/Ti and Al/N denote the Ti- and N-terminated surfaces, respectively. | Equilibrium | Adhesion energy | Removal energies | Material transfer
---|---|---|---|---
| interface distance [Å] | [eV/interface cell] | [eV/interface cell] | [Al layers]
(001) Al/N (top) | 2.06 | -0.61 | 1.16 | 0
(011) Al/N (bridge) | 1.39 | -2.09 | 1.35 | 2
(011) Al/Ti (top) | 2.77 | -0.73 | 1.35 | 0
(111) Al/Ti (hcp) | 2.22 | -1.78 | 0.80 | 2
(111) Al/Ti (top) | 2.67 | -0.94 | 0.80 | 1
(111) Al/N (hcp) | 1.04 | -1.90 | 0.80 | 2
(111) Al/N (top) | 1.87 | -1.38 | 0.80 | 1

For the Ti-terminated (111) surface orientation potential-energy curves are
presented in Fig. 9(a) for the two extremal alignments, Al/Ti (hcp) and Al/Ti
(top), which show the highest and lowest adhesion energies. As expected from
the energetics, material transfer occurs during separation, and both systems
end up in an energetically more favorable configuration compared to the
initial setup. In particular, one and two Al layer(s) for Al/Ti (top) and
Al/Ti (hcp), respectively, are transferred. This discrepancy in the number of
transferred layers cannot be explained from the energetics but could stem from
the different equilibrium interface distances. Compared to that of the hcp
alignment, this distance is significantly increased by almost 20% for the top
configuration, hindering the interaction between TiN and the subinterface Al
layer. For the Al/Ti (hcp) configuration snapshots of the structures during
approach and separation are presented in Fig. 10. During the approach at a
slab height of about 33.6 Å a large drop in interaction energy occurs for the
Al/Ti (hcp) alignment [see Fig. 9(a)] due to material transfer of the topmost
Al layer to the TiN slab [see Fig. 10(b)]. This is not the ground state since
a transfer of two layers would yield an even lower total energy. At this
distance, the transfer of the second Al layer is hindered by an energy barrier
of about $E_{b2}\approx$ 324 meV, which is significantly larger than for the
first layer alone, $E_{b1}\approx$ 122 meV. Upon further approaching, at a
slab height of about 32.8 Å, a slight kink occurs [see Fig. 9(a)] because the
Al slab is expanded into the space between the slabs [see Fig. 10(c)]. For a
further approach, the interaction energy follows an essentially parabolic
curve until the minimum energy is reached [see Figs. 9(a) and 10(d)]. The
subsequent separation is started from the equilibrium structure at the
interaction energy minimum. At first the red interaction energy curve lies on
top of the blue one [see Fig. 9(a)], meaning that the Al slab becomes extended
again [see Fig. 10(f)]. At a slab height of about 33.1 Å the two curves for
approach and separation start to deviate [see Fig. 9(a)] when the Al/TiN
compound separates [see Fig. 10(g)]. Two Al layers stick to the TiN slab and
form a stable configuration. This behavior during the complete loop is typical
for all cases exhibiting material transfer. While the Al slab is strongly
affected by the approach of the TiN slab, almost no changes in the TiN
structure are observed. The more pronounced impact on the Al slab is not
surprising when considering that TiN forms a much more rigid lattice than Al.
This claim is not entirely valid for the N-terminated (111) TiN slab, which
will be discussed in the following paragraph. Using the finally stable state
(TiN plus two Al layers) as a starting configuration for a new loop of
approach and separation versus an Al slab yields a reversible cycle. This
should be kept in mind when one is interpreting, for example, AFM experiments.
Upon the first contact between the tip and a particular spot on a surface
material transfer might occur, which in turn changes the contact properties
and forces between the tip and the surface. However, further encounters on the
same spot should then be within the reversible cycle and lead to the same
response.

(a)
33.8 Å

(b) 33.6 Å

(c) 32.8 Å

(d) 30.8 Å

(e)
30.8 Å

(f) 33.0 Å

(g) 33.1 Å

(h) 34.6 Å

Figure 10: (a)–(d) Approach and (e)–(h) separation of (111) Al and (111) TiN
slabs. Al, Ti and N are colored in red, blue, and green, respectively. (d) and
(e) show the structure at the relaxed equilibrium distance. The steps are
defined by the slab height.

The N-terminated (111) orientation is, in some respects, very similar to the
Ti-terminated one. As predicted, both configurations yield material transfer
for all tested alignments (see Fig. 9). However, as explained above, static
calculations completely fail to describe the equilibrium quantities of the
N-terminated case, whereas they result in good estimates for the other
orientations. This discrepancy is due to the behavior of the interfacial N
layer for N-terminated (111) TiN. In the absence of the counter Al slab, the
surface N layer is closely bound to the next Ti layer at a distance of 0.84 Å,
while in contact with an Al slab the distance grows to 1.47 Å at the
equilibrium configuration. This behavior is crucial for the energetics and can
be captured only when relaxations are included. The interfacial N layer is
actually closer to the next Al layer with a distance of 1.11 Å than to the
next Ti one. Due to this result the possibility of a diffusion of the
interfacial N layer into the Al slab was investigated. For all alignments with
the exception of Al/N (top), no energetically favorable configurations were
found. However, for the Al/N (top) alignment the exchange of the interfacial
Al and N layers and a subsequent relaxation of the system yield a favorable
state by about 683 meV, which is also about 235 meV lower than the previously
found minimum for the Al/N (hcp) alignment. In this favorable configuration an
Al-N-Al trilayer is formed, showing the wurtzite structure, which is typically
observed in aluminum nitride crystals. From the thermodynamical point of view
diffusion seems to be possible. Of course, for the full picture reaction paths
and energy barriers have to be considered.

The (011) surface orientation also presents an interesting case because due to
the energetic results [see Fig. 5(c)], material transfer is expected for only
the alignments Al/N (bridge), Al/N (top), and Al/TiN (hollow). As an example,
one can see from the loops given in Fig. 9(c) that Al/N (bridge) shows a
favorable configuration after separation corresponding to the transfer of two
Al layers, whereas the Al/Ti (top) case is reversible upon approach and
separation without any material transfer.

Finally, for the (001) surface orientation material transfer is not expected
for any of the alignments. Among all cases Al/N (top) has the largest adhesion
energy; therefore, if a material transfer occurs, it will happen for this
case. However, since the energy cost for the removal of an Al layer exceeds
the adhesion energy, no material transfer is observed [Fig. 9(d)]. The
deviation of the curves for approach and separation around 26.5 Å slab height
occurs due to the expansion of the Al slab upon separation until the interface
breaks apart and relaxes into the initial Al and TiN slabs.

In the literature some publications on tensile test simulations of Al/TiN
interfaces can be found, where the separation is achieved by increasing the
size of the whole simulation cell in one direction in discrete steps including
interim relaxations. Liu et al. Liu  _et al._ (2005) and Zhang et al. Zhang
_et al._ (2007) investigated the Al/TiN (111) and (001) interfaces,
respectively. Liu et al. obtained similar results with respect to material
transfer for the hcp alignment at the (111) surface for both terminations but
did not examine any further alignments of Al and TiN slabs at the interface.
Zhang et al. studied the Al/N (top) configuration of the (001) interface and,
in contrast to our work, found a material transfer of the top Al layer. This
discrepancy could stem from the different simulation approaches and
computational details. However, we repeated these calculations using a setup
for the separation of the slabs similar to that of Zhang et al. and did not
find any material transfer. Additional simulations for the different setups
used by Zhang et al. and in the present investigation testing the influence of
varying step sizes also did not lead to a material transfer.

###  III.4 Comparison of Surface Energies

The behavior of the different surface orientations can also be discussed from
the surface energy’s point of view. The surface energies for Al and TiN slabs
are presented in Table 3. It has to be noted that for (111) TiN the surface
energy depends on the termination and the chemical potential of nitrogen. Here
the lowest possible value for the surface energy is used, which is achieved by
the N-terminated surface at $\Delta\mu_{N}=0$ (see Fig. 2). The value of the
surface energy for the Ti-terminated surface at $\Delta\mu_{N}=0$ is about
three times larger, but its minimum is comparable to that of the N-terminated
case. For the (001) and (011) orientations the surface energy is independent
of the chemical potential Wang  _et al._ (2010). As shown in Table 3, the Al
surfaces always exhibit a smaller surface energy than the TiN ones. The
differences between Al and TiN are pronounced for the (011) and (111)
orientations. For (001), however, the surface energies are rather comparable.
Material transfer can be seen as a measure of surface-energy minimization by
creating a new energetically “cheap” surface and covering an “expensive” one
with it. This argument provides a hint about which surface orientations may
favor material transfer. However, for the full picture also other
contributions such as the interaction energy, which is influenced additionally
by the alignment of the slabs, also have to be considered. For example, the
surface-energy argument would suggest the possibility of material transfer for
(001) and cannot explain why only some (011) configurations exhibit this
feature.

Table 3:  Surface energies (in eV/Å2) of the Al and TiN slabs for the (001), (011), and (111) surface orientations. In the case of the (111) TiN surface the N-terminated one at $\Delta\mu_{N}=0$ is given here because it exhibits the lowest surface energy of all (111) TiN surfaces (see Fig. 2). | (001) | (011) | (111)
---|---|---|---
Al | 0.058 | 0.064 | 0.052
TiN | 0.087 | 0.174 | 0.094

###  III.5 Assessment of Computed Results

To validate the information presented above various additional tests were
performed. The results will be presented for the Ti-terminated (111) Al/Ti
(hcp) configuration. First of all, finite-size effects are a major concern.
Thus, the size of the simulation cell was increased laterally up to a
3$\times$3 surface cell and vertically up to a 19-layer Al slab. Also,
intermediate Al slab thicknesses were examined. The TiN slab was not extended
vertically because it is almost not affected by the approach of the Al slab.
In the case of laterally magnified simulation cells the number of k points was
decreased accordingly, e.g., for a 3$\times$3 surface cell, a
5$\times$5$\times$1 mesh was used. For all tested systems the equilibrium
interface distances, adhesion energies, and energy costs to remove Al layers
were found within about 2% of the values given above. The energies are
referenced to 1$\times$1 surface cells. Particularly, the results on material
transfer were not affected, meaning that the number of transferred Al layers
was not altered. For the 3$\times$3 surface cell the effect of fluctuations at
the surface was tested by moving one atom out of the surface plane at several
interface distances before material transfer occurs. These tests resulted in
the transfer of entire layers too since the shifted atom either relaxed back
into its originating slab or was transferred together with the rest of the
layer.

Furthermore, the effect of the chosen lattice parameters was investigated. The
simulations were repeated using the lattice constants of pure Al and TiN for
the lateral lattice parameters of the simulation cell. The equilibrium
interface distance again changed by only about 2%. Although the adhesion
energies were altered by about 4%, the removal energies were affected in a
similar way, resulting in the same material transfer. Moreover, the influence
of other approximations for the exchange correlation functional was tested as
already discussed for the removal energies above. The results are presented in
Fig. 11. The adhesion energies were enhanced similar to the removal energies,
again producing the same results for material transfer. Using the vdW
functional, the interaction between the slabs started at larger interface
distances. This behavior is expected because of the nonlocal correction added
in the vdW functional.

Figure 11: Comparison of calculated interaction energies for the Ti-terminated
(111) Al/Ti (hcp) configuration and various exchange correlation functionals,
namely, PBE, LDA, and opbB86b. The relaxed energies represent the separation
of the slabs starting from the equilibrium configuration. The differences
between static and relaxed curves at large heights occur due to material
transfer.

##  IV Conclusion and Outlook

Al/TiN interfaces were examined in detail by investigating the contact between
Al and TiN slabs showing low-index surface orientations within the framework
of density functional theory. Moreover, these contacts were established for
various lateral alignments of the slabs at the interface. It was shown that
interfacial properties such as the adhesion energy and the equilibrium
structure sensitively depend on the given configuration. This behavior can be
qualitatively explained by comparing the densities of state and the charge
densities of different configurations because distinct bond situations are
revealed at the interface. Furthermore, the approach and subsequent separation
of Al and TiN slabs were simulated to study the effect on the slabs,
especially the possibility of material transfer. The transfer of material from
an Al toward a TiN slab was observed for interfacial configurations, which
exhibited a larger adhesion energy than the energy cost to remove layers from
the Al slab. This is in agreement with the observation that metal-ceramic
interfaces break at the interface or in bulk areas according to their
interfacial adhesion Howe (1993b); Ernst (1995). The removal energy for Al
layers was found to depend on tensile or compressive stress. In all systems
showing material transfer one or two layers of Al stick to the TiN slab after
the separation and form an energetically favorable compound with respect to
the initial configuration. The differences in surface energies between the
slabs are not sufficient to explain the occurrence of material transfer
because the given alignment at the interface has to be considered as well. All
results were tested for various computational setups such as different sizes
of the investigated system or several approximations for the exchange
correlation functional. While properties such as the removal and adhesion
energies depend on these settings to some degree, the results for material
transfer are not affected.

The method used in this work can be, in principle, applied to any pair of
materials. However, complex materials or pairs with an unfavorable bulk
lattice mismatch may need very large simulation cells to be considered, which
means high computational demands, in order to preserve the translational
symmetry and to keep the distortions at an acceptable level. Furthermore,
larger cells also allow the inclusion of additional features. For example, the
distortions due to the lattice mismatch can be minimized, dislocations as well
as quasi-incommensurate contacts can be modeled, and even roughness could be
included to some degree, e.g., by using stepped surfaces or a regular grid of
small asperities.

## Acknowledgments

The authors thank G. Vorlaufer for fruitful discussions. G.F., M.W., P.O.B.,
P.M. and J.R. acknowledge the support by the Austrian Science Fund (FWF):
F4109 SFB ViCoM. Part of this work was funded by the the Austrian COMET-
Program (project K2 XTribology, No. 824187 and No. 849109) via the Austrian
Research Promotion Agency (FFG) and the Province of Niederösterreich,
Vorarlberg and Wien. This work has been carried out within the “Excellence
Center of Tribology” (AC2T research GmbH) and at the Vienna University of
Technology. Part of this work was supported by the European Cooperation in
Science and Technology (COST; Action MP1303). The authors also appreciate the
ample support of computer resources by the Vienna Scientific Cluster (VSC).
Figs. 1 and 3 were created employing VESTA Momma and Izumi (2011), Fig. 8
utilizing VisIt Childs  _et al._ (2012) and Fig. 10 using VMD Humphrey  _et
al._ (1996).

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