Bowker M., Davies P.R. (Eds.) Scanning Tunneling Microscopy in Surface Science, Nanoscience and Catalysis
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References
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95
4
Theory of Scanning Tunneling Microscopy and
Applications in Catalysis
Gilberto Teobaldi, Haiping Lin, and Werner Hofer
4.1
Catalysis and Scanning Tunneling Microscopy
Scanning tunneling microscopy (STM) has developed into one of the most ubiqui-
tous tools in surface science and the range of phenomena studied by this technique
is nowadays remarkably broad. These include surface topography, electronic and
vibrational properties, film growth, contact charging, molecular manipulation, and
many other phenomena ranging from the micrometer scale down to the subnan-
ometer scale. Concerning surface chemical reactivity, the use of STM and scanning
tunneling spectroscopy (STS) has proved fundamental for probing local surface
modifications associated with chemical reactivity at the solid–vacuum interface and
to follow, when possible, the mobility of reaction intermediates and final products
on surfaces in real time. Owing to the complexity of the studied systems in terms of
both surface morphology and variety of chemical reactivity, collaboration between
experiment and theory has emerged as a necessary tool in order to arrive at
sustainable models and a successful rationalization of experimental findings [1–4].
The aim of theory, when dealing with chemical reactivity at surfaces as imaged by
STM is twofold: on the one hand, simulations can provide insight into the energetic
landscape governing the chemical reaction from the reactants to the final products;
on the other hand, STM simulations are of utmost importance for an understanding
of the origins of image contrast and interpreting particular experimental images.
Modeling surface reactivity as imaged by STM – the first major challenge – is the
actual capacity of ab initio approaches to correctly describe the surface electronic
structure and the ensuing chemical reactants in the presence of other chemical
species dosed on the reaction substrate. Here, we consider the main practical
limitations that affect ab initio modeling of surface chemical reactivity such as the
need to work with simulated slabs of finite size and thickness and the approximated
description to exchange and correlation (XC) terms. In this chapter, we review for
a few selected applications how the modeled reactivity and the simulated STM
appearance of reactants and products are affected by the chosen trade-off between
the enforced level of approximations and the ensuing accuracy of the calculations.
j
97
4.2
Image Formation in an STM
The control parameter in an STM, the current in the tunneling junction, is always
due to the same physical process. An electron in one lead of the junction has a
nonvanishing probability to pass the potential barrier between the two sides and
to tunnel into the other lead. However, this process is highly influenced by (i) the
distance between the leads, (ii) the chemical composition of the surface and tip,
(iii) the electronic structure of both the systems, (iv) the chemical interactions
between the surface and the tip atoms, (v) the electrostatic interactions of the sample
and tip. The main problem, from a theoretical point of view, is that the order of
importance of all these effects depends generally on the distance and therefore on the
tunneling conditions [5–8].
Apart from the tunneling current, the other parameter required to define the
tunneling conditions (for a given tip) is the bias voltage, that is, the electrostatic
potential difference enforced between the probed surface and the probing tip. In
general, the change of bias voltage has two effects: (i) it shifts the Fermi level of one
lead (sample or tip) relative to the other lead and (ii) it compensates for this effect by
creating a surface dipole. Both of these effects are understood in principle, even
though no detailed simulations actually exist. The reasonis that a simulation requires
treating the coupled system of two leads and the interaction range in a nonequilib-
rium situation. Even though the theoretical framework for such a treatment in the
STM context, the nonequilibrium Green function formalism of Keldysh, has been
known for some 40 years [9], its numerical implementation became possible only
recently [10]. Within an STM, the method is generally not applicable due to the
different lateral symmetry of surface and tip systems. However, provided the
electronic density of states (DOS) of the surface and tip structure are fairly smooth,
the current increases approximately linearly with the applied bias voltage. This is
due to an increase in the energy difference between the Fermi level of the sample
and the tip, which increases the number of electron states provided by one lead, the
number of empty states provided by the other, and to the condition of resonant
tunneling.
The differences in Fermi level due to bias voltage, combined with the general
differences between the Fermi level of a sample and the tip, introduces electrostatic
effects into the interfaces. A thorough theoretical investigation of this issue for
differently conductive substrates, carried out at different level of theory [11–14] has
pointed out that although electrostatic interactions are present between the tip and
the substrate, for actual measurements under normal experimental conditions the
effect is not strong enough to affect the STM image in a substantial way [5].
Unlike electrostatic forces, chemical forces between the probing tip and the probed
surface have been shown to profoundly affect the tunneling current from a certain
onset. Owing to the advent of first-principle methods and powerful computers, it
could finally be resolved by acalculation of the combined tip– sample system [15]. The
point of onset for chemical bonding on metals was found to be at a distance of 4–5Å.
As the tip approaches the surface, chemical forces rapidly become large enough to
98
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4 Theory of Scanning Tunneling Microscopy and Applications in Catalysis
destroy the stability of a tip–sample system in an STM due to ions jumping from the
tip to the surface and vice versa. The point of destruction is at about 4 Å. From the
onset of chemical forces to the limit of mechanical stability, the two main effects are
(i) an increase in the tunneling current beyond exponential growth and (ii) an
increase in surface corrugation by a factor of 2–3. Both effects are observed in
experiments [16, 17]. They show, unambiguously, that high-resolution STM operates
within the range of chemical forces, and that the main effect, omitted in any
straightforward perturbation model, is the relaxation of surface atoms due to
chemical interactions.
Modeling STM images involves several stages: (i) establishing a realistic model of
the surface from experiment and theory; (ii) establishing a realistic model of an STM
tip using properties known experimentally and inferred from theory; (iii) explicitly
modeling the interactions between the tip and the surface; (iv) calculating the
tunneling current for the applied bias; (v) comparing with the experimental data.
A fundamental requirement of any STM simulation is the demand that the electronic
states of the surface, and ideally of the tip, is precisely simulated because these states
determine the obtainable contrast and ultimately the simulated STM image. Here,
the level of accuracy of the simulation critically depends on the accuracy of the
simulation in modeling both the energetic localization of the electronic states with
respect to the surface (tip) Fermi level (i.e., chemical potential at 0 K) and the real
space distribution outside the slab, that is, their decay in the vacuum region. The first
stage of the modeling process is, thus, establishing a good representation of the
surface being studied. We note that on the basis of recent results [3, 18–20], an
accurate description of the energetics associated with a given reactivity scenario
and/or the electronic structure of the reaction intermediates may not be sufficient to
accurately account for the experimentally detected STM appearance as will be shortly
reviewed.
Concerning the probe tip, its correct description is indeed another key factor
determining the final level of agreement with the experiment. Owing to the
complexity of the subject, we refer the reader to detailed reviews existing on the
subject [5, 6]. Here, we only note that since no experimental atomic information is
ultimately known about the probing tip to date, the most accurate solution of the
problem relies on empirical comparison of modeled images simulated with different
tips and available experimental data. In this perspective, the recently reported
experimental approaches toward simultaneous recording of STM and atomic force
microscopy (AFM) images represent an extremely valuable method of comparison
for explicit modeling of the tip influence on the image and in turn the identification
of a library of most likely tip geometries for certain experimental conditions [4].
4.3
Simulating Tunneling Currents
A complete survey of all methods to calculate the tunneling current, developed over
the past few decades, is given in Ref. [21]. In increasing order of theoretical difficulty,
4.3 Simulating Tunneling Currents
j
99
the four main approaches are (i) the Tersoff–Hamann approach [22, 23], where
constant current contours are modeled from the electronic structure of the surface
alone; (ii) the transfer Hamiltonian or the Bardeen approach [24], where the tip
electronic structure is explicitly included in the calculation; (iii) the scattering or
Landauer–B
€
utticker approach [25], which includes multiple pathways of tunneling
electrons from their initial to their final crystal states; and (iv) the Keldysh or
nonequilibrium Greens function approach [26], which also considers inelastic effects
such as electron–electron or electron–phonon scattering. For technical details of
the various levels of approximation involved, the reader is referred to the literature
[28–33]. In the following section, we shall base STM simulations mainly on the
multiple scattering approach developed to first order [27].
4.4
Simulating Chemical Reactivity
Disregarding the implementation details with respect to the wave function approx-
imation [34], two critical aspects when modeling the chemical reactivity at surfaces
are (i) the evaluation of the XC-terms and (ii) the use of simulation cells of finite size.
With respect to (i) we limit the review to a limited number of studies addressing the
role of the XC description with respect to the modeled electronic structure and the
ensuing simulated STM appearance for a given system (see below and Refs [3, 20]).
The main reason for the small numbers of such studies is the remarkably high
computational cost associated with the explicit inclusion of Hartree–Fock (HF) terms
in the description of XC potentials [35, 36]. This increased computational cost
inevitably demands larger computational facilities that at the moment are not
generally available for academic research. So far, the most notable improvements
by enhanced XC descriptions (as those provided by hybrid DFT approaches such as
B3LYP [35] or HSE03 [36]) have been found for ionic semiconductors [37–40].
Unfortunately, the larger computational requirements associated with hybrid-DFT
approaches is hardly compatible with another aspect characterizing ionic semicon-
ductors, that is, the role of long-range electrostatic effects. In fact, for these substrates,
the use of large simulation cells and thick slabs is a fundamental requirement to
obtain converged results about basic reactivity indicators, for instance, the adsorption
energy [3, 41, 42]. Just to quote an example, it has been reported that the adsorption
energy of molecularly adsorbed O
2
on hydroxylated TiO
2
(1 1 0) [41] or H
2
O on clean
TiO
2
(1 1 0) [1, 3, 42] dramatically depends on the simulated coverage (see Figure 4.1).
Thus, the advantages of an improved XC description might be balanced by
the simulation biases associated with the reduction of the size for the simulation
cell or, equivalently, with an overestimated coverage compared to the actual situation
imaged in the experiments. This aspect is even more delicate for simulations
aiming at assessing the relative stability of different intermediates characterized by
different long-range interactions (for instance, O
2
-related products on hydroxylated
TiO
2
(1 1 0) [43]). In fact, within the same modeled extent of the simulation cell, in
principle, nothing prevents this finite size bias changing as the polarity of the
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4 Theory of Scanning Tunneling Microscopy and Applications in Catalysis
intermediates change, negatively affecting the reliability of final computed stabilities.
In our view, this aspect may represent one of the main (indirect) limitations of the
use of hybrid-DFT approaches in modeling the chemical reactivity of ionic semi-
conductors. Extra care should, therefore, be taken when choosing the simulated
system size.
4.5
Catalytic Water Production
Catalytic oxidation of hydrogen on transition metal surfaces plays a crucial role in
various subjects such as electrochemistry or heterogeneous catalysis, and it has
important applications, for example, for making hydrogen fuel cells. The first report
of this phenomenon dates back to the early nineteenth century, when hydrogen was
observed to react with oxygen at room temperature to produce water on a Pt surface.
The remarkable ease of the reaction led to the definition of the term catalysis [44].
In the next 100 years, these reactions were intensively investigated on various metals,
such as silver, gold, rhodium, and palladium [45–49]. In these studies, the exper-
imental techniques limited the observations to the macroscopic or mesoscopic level.
Although it was found that the catalytic production of water from hydrogen and
oxygen is always accompanied by the simultaneous oxidation of the metal surface,
the proposed reaction mechanisms could not be firmly proven [50, 51]. Since 1982,
the STM has been offering the possibility of direct real-space determination of
surface structures at the microscopic scale [52, 53]. Over the past two decades, the
1/8 1/4 1/2 3/4 1
OH Covera
g
e (ML)
perfect
OH
O Vacancy
2.2
2.0
1.8
1.6
1.4
1.2
1.0
0.8
0.6
0.4
0.2
0.0
Adsorption Energies (eV)
Figure 4.1 Calculated O
2
adsorption energies with respect to OH
coverages. For comparison, the O
2
adsorption energies on the
perfect surface and in the vicinity of the O vacancy are also shown.
(Reprinted with permission from Ref. [41].)
4.5 Catalytic Water Production
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101
reaction transition states, that is, the formation of hydrogen bonding networks have
been repeatedly imaged in a number of STM measurements [54–56]. However, since
STM images are determined by the electronic states rather than the atomic positions
at surfaces, the chemical identification of the intermediate states remained some-
what unclear. Thus, only DFTcalculations and STM simulations in comparison with
experimental data can attain a complete and atomic-scale understanding of the
reaction mechanism of catalytic water production on transition metal surfaces.
Recently, STM measurements of a water production reaction were undertaken on
a well-designed (2 1)p2mg-O/Rh(1 1 0) surface under ultra high vacuum (UHV)
conditions [18]. In the initial structure, a Rh(1 1 0) surface is covered by an oxygen
overlayer, in which O atoms adsorb in a zigzag fashion at the threefold coordinated
sites. When the surface is exposed to molecular hydrogen, the reaction propagates
on the whole surface in a wave front, after a short induction time. The whole process
can be characterized by three distinct steps: in the first step, half of the oxygen atoms
are vaporized, the Rh surface is gradually covered by paired features, and the whole
surface shows a c(2 4) symmetry (see Figure 4.2a);in the second step, a close packed
c(2 2) island shows up and grows at the expense of the c(2 4) structure; it expands
to about half the surface. In the third step, this c(2 2) island gradually shrinks and
finally disappears, leaving behind a clean Rh(1 1 0) surface.
The interpretation of the two intermediate structures, that is, the c(2 4) and the
c(2 2) structures, is obviously of prime importance for an understanding of the
mechanism of this catalytic oxidation reaction. A detailed investigation of the c(2 4)
Figure 4.2 Intermediate stage of the reaction with c(2 4)-O
structure on Rh(1 1 0): (a) zoom on the surface behind the
reaction front. Dimensions: 11.5 nm 7.5 nm. I ¼1 nA,
V
B
¼þ0.13 V. (b) STM image after dosing 0.6 l oxygen and
annealing at 270 K. Dimensions: 20 nm 20 nm. I ¼1 nA,
V
B
¼þ0.42 V. (c) Structural model of the c(2 4)-O.
(d) Simulated STM image (V
B
¼þ0.13 V, I ¼0.05 nA).
102
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4 Theory of Scanning Tunneling Microscopy and Applications in Catalysis
structure shows that the height of the paired features is 0.3–0.4 Å, with an internal
distance of about 2.77 Å. All these values are in agreement with the experimental
observation in Ref. [57], where, on a Rh(1 1 0) surface at low coverage, pairs of oxygen
atoms align along the [1
1 0] direction, the internal distance varying from 3.3 Å (at the
lowest coverage) to 2.5 Å (at the highest coverage). The c(2 4) paired structures are,
thus, very likely due to the presence of adsorbed oxygen atoms. On the basis of this
assumption, first-principle calculations were performed to determine the ground-
state adsorption configuration. The structure optimization is carried out with the
Vienna ab initio simulation package (VASP) [58–61], a density functional theory-
based code for systems with periodic boundary conditions. The ion–electron inter-
actions are described by the projector augmented wave (PAW) [62] method and the
exchange correlation potentials are calculated by the generalized gradient approx-
imation (GGA) of Perdew and Wang (PW91) [63]. The metal surface is modeled by
a supercell containing a five-layer slab, in which the top two layers of Rh atoms are
allowed to relax in three dimensions; the vacuum range between the slabs is about
19 Å. In total, six configurations with c(2 4) symmetry are analyzed for the structure
search. The lowest energy adsorption sites are found to be short-bridge sites, in which
the adsorption energy per O atom is at least 0.1 eV larger than for the models. This
structure model is shown in Figure 4.2c. However, we note that the OO distance in
a pair is about 3.8 Å rather than the 2.7 Å observed in the STM experiments. In
general, this can be the result of a contaminated tip, strong tip–adsorbate interaction
and corresponding atomic relaxations, or electronic effects of the reacted surface.
If the tip is contaminated, its apex is most likely attached to a hydrogen molecule or
H atoms. As a consequence, the conductance of this tip should be much lower than
that of a clean tungsten tip. Since this conductance change has not been reported,
it can be concluded that the reduced OO distance is not the effect of a contaminated
tip. Surface–tip interactions are evaluated by calculating the interaction between
the reacted surface and a tungsten cluster at low distance. Here, the calculations
indicate that there is no substantial relaxation due to interactions between the two
leads. Consequently, the only possibility left is that the electronic surface structure
somehow changes the appearance of the oxygen positions.
Subsequently, we calculated constant current contours using the multiple scat-
tering approach and a clean tungsten tip. The results of these simulations are
shown in Figure 4.2. It turns out that although the actual distance between oxygen
atoms is 3.8 Å, the imaged distance is substantially smaller at 2.7 Å. The reason for
this discrepancy is the large size (more than 4 Å) of the tip apex: in this case the
contributions from the two electronic states at the two oxygen atoms are not
completely separate, as the tip scans across the surface, and the protrusions,
therefore, appear to be closer together than in reality. Clearly, this is a generic effect
of the surface electronic structure that should also be present in other experiments
and thus quite general. The agreement of experimental results and theoretical
simulations suggests that the c(2 4) structure is formed by the oxygen atoms at
the short bridge site of the Rh(1 1 0) surface.
After dosing hydrogen, a different structure with c(2 2) symmetry is formed.
The more condensed c(2 2) island is characterized by a hexagonal arrangement of
4.5 Catalytic Water Production
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