Surface Thermodynamics and Vibrational Entropy

Abstract

As with bulk material, a few thermodynamic variables determine the stability and equilibrium properties of solid surfaces. Since the environment at the surface is typically very different from that in the bulk, as a result of the lack of symmetry created by the presence of the surface, thermodynamics plays an important role in determining surface structure and dynamics. The quantity of interest here is the surface free energy, which inherently includes the contribution of vibrational (and configurational) entropy. Given the existence of surfaces vibrational modes whose features are distinct from those in the bulk, and dependent on local surface geometry and electronic structure, the emphasis in this chapter is on the characteristics of vibrational entropy, which, in turn, affect surface thermodynamical quantities that are in excess of values in the bulk. Special attention is paid to characteristics of vibrational density of states of low and high Miller index surfaces and their contribution to vibrational entropy, and, hence, to thermodynamical functions. In fact, it is argued that the distinguishing features in the vibrational density of states, namely enhancement of the number of modes at low frequencies and appearance of modes above the bulk band, highlight the impact of the undercoordinated atoms at surfaces, steps, and kink sites and lead to variations in the local surface electronic structure. The characteristics found on high Miller index surfaces in particular pave the way for understanding vibrational dynamics of nanoparticles. Contact is made with experimental data where available.

In this chapter, after presenting a summary of bulk thermodynamics in Sect. 3.1, the stage is set for summarizing the essentials of surface thermodynamical functions in Sect. 3.2. Surface nomenclature is discussed in Sect. 3.3 and details of theoretical techniques are presented in Sect. 3.4. A sampling of results obtained on a few surfaces is provided in Sect. 3.5 and a summary is presented in Sect. 3.6.

Over four decades of experimental and theoretical research in surface science has brought us to the point where we can probe the subtle role played by thermodynamics in determining unique features of solid surfaces and nanostructures. The lack of symmetry created by removal of the top half-atoms to create a surface inherently leads to a heterogeneous environment with nonuniform local geometric and modified electronic structure that can promote a plethora of surface phenomena, which continue to be revealed at length scales ranging from the microscopic (localized surface states and vibrational modes), to the macroscopic (surface tension, surface acoustic wave) and at time scales going down to femtoseconds. In this Handbook, a number of chapters address the experimental and theoretical findings of a multitude of surface phenomena, allowing this chapter to concentrate only on the peculiarities of surface thermodynamics that make the surface an interesting playground for realization of physical and chemical properties than that in the bulk.

In considerations of relative stability of surfaces, nanostructures, as well as bulk phases of different crystallographic orientations, the quantity of interest is the free energy, which includes contributions from the structural potential energy and the system’s vibrational and configurational entropy. Assuming the structural potential energy to be insensitive to temperature variations, it is the entropic contributions that control the surface phase diagram and structural stability. In this regard, configurational entropy is an important constituent for any system containing more than one type of element (for example, alloys), but for surfaces and nanostructures of single elements, it is vibrational entropy that accounts for the temperature dependencies of surface free energy, mean-square vibrational amplitudes of surface atoms, the surface Debye temperature, and the surface heat capacity, albeit within the harmonic and quasiharmonic approximation of lattice dynamics. It is also the quantity that may determine the equilibrium shape of crystal surfaces and its possible structural phase transitions and surface reconstructions. These comments are not to disregard the role of configurational entropy for single-element surfaces at temperatures that accompany structural disorder. Such considerations are beyond the scope of this chapter. Knowledge of surface free energy, together with that of the step and kink free energy, is also essential for considerations of surface faceting, bunching, and roughening. The extraction of free energy from experimental data is, however, nontrivial [3.1]. The lattice contribution, which can be critical for determining structural transitions, is nonzero, albeit a small fraction of the structural energy. It is, thus, encouraging to see the flurry of activity in analyzing the contribution of vibrational entropy [3.2] to the thermodynamic functions for several surface systems [3.3, 3.4, 3.5]. These calculations have already provided a qualitative measure of the effect of vibrational entropy on surface stability and structure and have set the stage for a systematic evaluation of the local vibrational contribution to the free energy. Since these calculations were based on the usage of many-body interaction potentials [3.6], questions have been asked about their accuracy, particularly for 5d metals Pt, Ir, and Au, for which these potentials are not expected to work as well as they do for Ag, Cu, and Ni. With the availability of ab-initio electronic structure methods based on the density functional perturbation theory [3.7, 3.8], surface phonon dispersion curves can be calculated with remarkable accuracy [3.9]. These dispersion curves further lend themselves to the extraction of vibrational density of states and, thus, of the vibrational contribution to surface free energy and entropy. Efforts have thus been made to analyze surface thermodynamic properties of the several metals using ab-initio methods and to compare the findings with those obtained using semiempirical approaches [3.10].

1 Some Essentials of Bulk Thermodynamics

In this section, some background information is provided on thermodynamic functions to provide context and facilitate discussion, greater details of which can be found in textbooks [3.11, 3.12]. With the entropy already mentioned above, two extensive thermodynamic variables need to be introduced: the Helmholtz free energy F equivalent to the maximum amount of work a system can do at constant volume and temperature and the Gibbs free energy G, which is the maximum amount of work a system can do at constant pressure and temperature; G is a minimum for closed systems at equilibrium with a well-defined temperature and pressure. Both Helmholtz and Gibbs free energies are known as thermodynamic potential. Furthermore, the first law of thermodynamics states that the increase in energy dU of a system is equal to the difference of the heat absorbed by the system δQ and the amount of work done by the system δW as quantified by

$$\mathrm{d}U=\updelta Q-\updelta W\;.$$

(3.1)

Equation (3.1) implies that there is a quantity U called the internal energy which is a function of the state of the system (temperature and pressure) and that the difference between two values of U is independent of the path of getting from one state to the other. In other words, U is the integral of an exact differential dU. If the energy of the system can be described by the thermodynamic variables' pressure, temperature, and volume, then for an infinitesimal quasistatic reversible process, the work done by the system is PdV, and the heat absorbed is TdS. Thus,

$$\mathrm{d}U=T\mathrm{d}S-P\mathrm{d}V\;,$$

(3.2)

from which temperature and pressure follow as respective derivatives of the internal energy U. Since S and V are often inconvenient independent variables [3.13], it is more usual to work with T and P. The thermodynamic potentials, F and G introduced before and the enthalpy H are then defined by

$$\begin{aligned}F & =U-TS=U-S\left(\frac{\updelta U}{\updelta S}\right)_{v}\;,\end{aligned}$$

(3.3)

$$\begin{aligned}H & =U+PV\;,\end{aligned}$$

(3.4)

$$\begin{aligned}G & =U-TS+PV=H-TS\;.\end{aligned}$$

(3.5)

While the quantities form the basis of thermodynamical description of bulk material with an infinite sea of particles, Gibbs’ introduction [3.14] of the explicit dependence of U on the number of particles N and their related chemical potential μ allows us to extend these thermodynamic concepts to solid surfaces. There is, thus, a consensus that in equilibrium, a one-component system can be fully defined by its internal energy (U), which is a unique function of the entropy (S), volume (V), and number of particles (N), such that

$$\begin{aligned}U & =U(S,V,N)\;,\end{aligned}$$

(3.6)

$$\begin{aligned}\mathrm{d}U & =T\mathrm{d}S-P\mathrm{d}V+\mu\mathrm{d}N\;.\end{aligned}$$

(3.7)

The reader is referred to more detailed analysis in Zangwill [3.11] from where some of the description has been extracted.

2 Surface Thermodynamic Functions

When it comes to surfaces, in addition to volume V, area (A) should also play a role, since we need energy to create a surface with a particular surface energy. The creation of a surface should inherently involve an increase in the total internal energy proportional to the area. Thus, the internal energy equation should include an additional term

$$U=TS-PV+\mu N+\gamma A\;,$$

(3.8)

where γ is the surface tension. Of course, the introduction of the surface brings in an ambiguity, as the boundary of the surface is not well defined. For example, the density of the electrons in a one-component system will not fall off sharply at the surface. Rather, it would tail off as one moves away from the surface, in a manner characteristic of the material. Additionally, surface thermodynamical quantities would follow from those calculated for the bulk with the inclusion of one term that accounts for excesses from the bulk value needed to create the surface. Equation (3.8) has also introduced another observable, surface tension, which is a macroscopic property of a material.

Surface internal energy may, thus, be defined as an excess over the bulk value in the manner described very nicely by Somorjai and Li [3.12], assuming that a crystalline solid is bound by two surfaces and denoting the energy and entropy of each bulk atom by U^b and S^b, respectively, and the surface energy per unit area as U^s. The total internal energy of the solid is then

$$U=NU^{\mathrm{b}}+AU^{\mathrm{s}}\;.$$

(3.9)

Here, AU^s is the excess from the bulk. In the same way, the total entropy of the system can be written as

$$S=NS^{\mathrm{b}}+AS^{\mathrm{s}}\;.$$

(3.10)

Note that the surface term (with the suffix s) is with respect to per unit surface area. Bearing in mind that the surface terms are excess over the bulk value in the manner described above, we can write the expressions for the surface Helmholtz and Gibbs free energies (per unit area) and the enthalpy (per unit area) in terms of the corresponding expressions for the bulk

$$\begin{aligned}F^{\mathrm{s}} & =U^{\mathrm{s}}-TS^{\mathrm{s}}\;,\end{aligned}$$

(3.11)

$$\begin{aligned}G^{\mathrm{s}} & =H^{\mathrm{s}}-TS^{\mathrm{s}}\;.\end{aligned}$$

(3.12)

Here, H^s is the specific surface enthalpy, which is the excess heat absorbed by the system per unit surface area created under constant external force, i. e., pressure. By the same token, the total Gibbs free energy of the system can be written in terms of G^b, the Gibbs free energy per bulk atom and G^s, and the Gibbs free energy per unit surface area

$$G=NG^{\mathrm{b}}+AG^{\mathrm{s}}\;.$$

(3.13)

We next move on to considerations of some of the thermodynamic functions for surfaces and other reduced dimensional materials. However, before we do so we need to establish the nomenclature, the calculation methodology, and provide some background information.

3 Surface Nomenclature and Geometry

Bulk solids exist in a number of crystallographic structures, details of which can be found in standard textbooks [3.15]. The fundamental property of a solid is its inherent periodic structure, which is defined by a lattice containing an infinite array of points generated by a primitive lattice translation vector

$$\boldsymbol{T}_{m}=m_{1}\boldsymbol{a}+m_{2}\boldsymbol{b}+m_{3}\boldsymbol{c}\;,$$

(3.14)

where m_i is an integer (positive, negative, or zero) and a, b, and c are three mutually perpendicular vectors, generally taken to be along the Cartesian x, y, and z coordinates. A lattice point m₁a + m₂b + m₃c lies in the direction [m₁ , m₂ , m₃]. Generally, the values of m_i are divided by their greatest common divisor, so as to have the smallest integer set. For example, the body diagonal of a unit cell is denoted by the direction [111] (rather than [333], etc.). Furthermore, a set of directions, such as [100], [010], [001], $[\bar{1}00]$, $[0\bar{1}0]$, $[00\bar{1}]$, which may be related by symmetry, are denoted by the bracket ⟨100⟩.

The planes containing the lattice point define the crystal structure and the ensuing symmetry and other geometrical characteristics of the surface. These planes are usually defined in terms of Miller indices (hkl), where the values of h, k, and l follow from the following. If a plane intersects the axes (m₁a, m₂b, and m₃c), the Miller indices of the plane are the set of integers, without a common multiple, that are inversely proportional to the intercept of the axes with the plane. That is, $h:k:l=1/m_{1}:1/m_{2}:1/m_{3}$. Thus, in a cubic lattice, the direction ⟨hkl⟩ is perpendicular to the plane (hkl). Here, too, a family of planes that are equivalent by symmetry are denoted by curly brackets { }. These Miller indices provide a convenient way to classify surfaces with respect to the extent of their terraces and regularly present steps and kinks. As will be seen below and was already presented in [3.5], the low and high Miller index surfaces provide two sets of surfaces with distinct crystallography.

3.1 Low Miller Index Surfaces

The low Miller index surfaces are those for which h, k, and l have the values 0 or 1. These are extended surfaces without any step or kink [3.16]. A simple example is that of a cube with its six faces ⟨100⟩, ⟨010⟩, etc., represented collectively by {100}. The low Miller index surfaces (100), (110), and (111) of both fcc and bcc cubic crystals are shown in Fig. 3.1. Note that the (111) surface is the most close-packed, while the least so is the (110) surface. In the early days of surface science, the majority of investigations were carried out on the three low Miller index surfaces for obvious reasons. They were relatively easy to characterize experimentally with techniques such as low-energy electron diffraction (LEED) and were amenable to theoretical calculations, both analytical and numerical with the then available computational power. However, the larger number of low-coordinated sites offered by surfaces containing steps and kinks compared to those without them, led both experimentalists and theorists to venture into the more complex, and intriguing, local environment offered by the high-Miller index surfaces, also called vicinal surfaces. These surfaces can now be created and characterized with a variety of techniques, as summarized in several other chapters in this Springer Handbook. Their well-defined periodic arrangement of steps and kinks provides regions of low coordination with interesting implications for their physics and chemistry, as we shall see in this book. They also serve as benchmarks for understanding properties of nanoparticles while offering sites with a range of low coordination that lack the periodicity of vicinal surfaces.

Fig. 3.1

Low Miller index surfaces

3.2 High Miller Index Surfaces

The simplest vicinal surfaces consist of regularly-spaced low Miller index terraces separated by monatomic steps. A slight miscut of the crystal at an angle slightly off the low Miller index ((100), (111), and (110)) planes will give rise to a vicinal surface. In Fig. 3.2, we depict a hard sphere diagram of an fcc(977) surface showing the geometry of the terrace and the step face. On this surface, the terrace has fcc(111) orientation and is eight-atom wide, while the step face has a (100) microfacet. This vicinal is created by cutting the crystal at 7^∘ away from the (111) plane towards the $[2\bar{1}\bar{1}]$ direction, which results in monatomic steps along the $[0\bar{1}1]$ direction. Note that on the fcc(111) surface, the ⟨110⟩ direction is not parallel to any plane of symmetry, and there exist two different ways of generating monatomic stepped surfaces: by directing the miscut angle towards either the $[2\bar{1}\bar{1}]$ or the $[\bar{2}11]$ direction. In the case of the former, as we can see in Fig. 3.1, the step face has a (100) microfacet (the so-called A type), while in the case of the latter, it would yield a (111) microfacet (the so-called B type). Such a surface with an eight-atom wide fcc(111) terrace and a (111) microfacet is the fcc(997) surface for which the miscut angle is 6.5^∘. It is complementary to the fcc(977) surface shown in Fig. 3.2. Since a monoatomic stepped surface may have a (100) or (111) microfacet, a more self-contained notation for the surface structure was introduced by Lang et al [3.18] in the general form S(h_tk_tl_t) × (n_sk_sl_s), where (h_tk_tl_t) represents the Miller indices of the terrace plane and (n_sk_sl_s) that of the step face, and S is the atomic width of the terrace. In Table 3.1, we summarize the notation for the various vicinal surfaces that are discussed in this chapter, together with their miscut angles θ and the interatomic separations for ideal (bulk terminated) geometry. Here, w is the width of the terrace (Fig. 3.2) and f .  . r is the registry (Fig. 3.5).

Fig. 3.2

Hard sphere model of the fcc(977) surface that is a vicinal of fcc(111) with an eight-atom wide terrace and a monatomic (100) microfaceted step face. The miscut angle θ and the terrace width w are shown in the inset. (Adapted from [3.17], © IOP Publishing. Reproduced with permission. All rights reserved)

Table 3.1 Structural notation and geometric features for several fcc vicinals. Here, d_b is the bulk interlayer separation, Δr is the surface registry and w is the terrace width (all in units of a, the lattice constant). (Adapted from [3.17], © 2003 IOP Publishing)

This relationship of the vicinal surface to the low Miller index mother surface is also nicely captured in Fig. 3.3, which presents a stereographic view of stepped surfaces of varying terrace widths and geometries, drawn around the pole at fcc(100) and converging on the other two low Miller index surfaces (111) and (100). As already mentioned, the two equal terrace-width vicinals of fcc(111), fcc(997), and fcc(977) fall on the two sides of the graph around (111) separated by 60^∘. Similar to the (997) surface, the (331) and (551) surfaces have a B-type step face but only three-atom wide terraces.

Fig. 3.3

A stereographic projection for fcc crystals around the (100) pole showing several vicinal surfaces in relation to the low Miller index surfaces from which they are derived. (Adapted from [3.17], © IOP Publishing. Reproduced with permission. All rights reserved)

For fcc(100) surfaces, the stereograph map neatly separates the two types of step face of its vicinals: those with a close-packed (111) microfacet lie along the line joining the (100) pole to (111) and those with the (110) microfacet along the other line. An angle of 45^∘ separates the in-plane orientation of these steps. Similarly, the set of loosely and close-packed step faces of vicinals of fcc(110) flank the location of (110) on the stereograph with a separation of 90^∘ between them. Thus, along each side of the stereograph the terrace geometry and width change logically while maintaining the same geometry for the step face. In the Miller index notation, vicinals lying along the (100)–(110) side of the stereograph have indices (hk0), while those along (110)–(111) have (hhk) and those along (111)–(100) are denoted by (hkk).

To examine the finer details of the local atomic environment, we take the example of a set of vicinal surfaces whose terraces are approximately three-atom wide with either (100), (111), or (110) crystallographic orientation and two types of step microfacets. These six surfaces are shown in Fig. 3.4.

Fig. 3.4

A top view of six vicinals of three-atom wide terraces. (Adapted from [3.17], © IOP Publishing. Reproduced with permission. All rights reserved)

Here, (511) and (310) are vicinals of fcc(100), with (111) and (100) microfaceted step faces, respectively. As is noticeable in Fig. 3.4, the (310) surface has an open-step arrangement, akin to a regularly kinked step edge. The coordination of the step atoms on the fcc(310) is also 6, which is typical of atoms at a kink site. Similarly, the (211) and (331) surfaces in Fig. 3.4 are corresponding vicinals of the fcc(111) surfaces with the A and B types of step microfacet, respectively. Finally, the corresponding vicinals of the fcc(110) surface are the (551) and (320) surfaces, the former with a close-packed (111) step face and the latter with a kinked (100) microfacet. The structural parameters of the above six surfaces, given in terms of the lattice constant, in Table 3.1, together with those of few other related vicinal surfaces, show that the fcc(977) surface is just a broader terrace version of the fcc(211) surface, while the fcc(410) surface is related to the fcc(310) surface, except for the difference in terrace width.

As in several previous studies [3.11, 3.12, 3.14, 3.4] we will use the following notation for the surface atoms, as illustrated in Fig. 3.5: SC for atoms in the step chain; TC for atoms in the terrace chain (if there are more than one, we label them as TC1, TC2, etc.), and CC for atoms in the corner chain. For the three surfaces (310), (320), and (551), there are actually five undercoordinated sites—three in the top terrace and two underneath. On the other hand, on the (511), (211), and (331) surfaces, there are only three undercoordinated sites. Together, these six surfaces provide atoms with the range of coordination extending from 6 to 11. As we shall see the atom labeled BNN in Fig. 3.5, a bulk nearest neighbor of the corner atom contributes interestingly to the vibrational density of states of these surfaces. Note that in the calculations,the x and y-axes are taken to lie in the surface plane, the x-axis being perpendicular to the step and the y-axis along the step. The z-axis is along the normal to the surface.

Fig. 3.5

A side view of the vicinal surface showing the step (SC), terrace (TC), corner (CC) and BNN atoms, interlayer separations d_ij, and surface registry (Δr). (Adapted from [3.17], © IOP Publishing. Reproduced with permission. All rights reserved)

4 Theoretical Techniques

The goal in this section is to provide the reader with some details of theoretical techniques that have been used to calculate the ingredients mentioned above that are needed to determine surface thermodynamical functions such as vibrational entropy, heat capacity, mean-square vibrational amplitudes, and, subsequently, the Debye temperature for the surface in question. We put Debye in italics here as the concept of Debye temperature, which is based on the Debye model for the vibrational density of states, is not valid for surfaces and nanostructures for which, as we shall see, the vibrational density of states does not follow the same dependence on energy as the atoms in the bulk [3.19]. Of course, the quantity of prime importance is the vibrational density of states to calculate which one needs to first evaluate the system minimum energy configuration, followed by a calculation of the system vibrational dynamics and, thence, that of the thermodynamical functions. Calculations for details of the electronic structure, charge redistributions, and local electronic densities of states are also of interest, but as they are included in other chapters in this book, they will not be discussed here in any detail. In this section, we provide some of the essentials for the calculations of:

• a) Geometrical structure

• b) Vibrational dynamics

• c) Vibrational free energy.

4.1 Determination of Surface Structure and Energetics

Once the coordinates of atoms for the system with the surface geometry discussed in Table 3.1 are generated, it is necessary to allow the system to relax so that atoms find their equilibrium positions in their undercoordinated sites. As is well known, atoms in the surface layers of a solid relax away from their bulk terminated positions, as a result of the bonds that were severed in the creation of the surface. These relaxations can be inwards or outwards with respect to the surface plane. To obtain the relaxed position standard algorithms, such as simulated annealing, steepest descent, and conjugate gradient methods, are used for minimizing the total energy of the system [3.20]. A critical input for all such calculations is the form of the interaction between the atoms/ion cores/electrons, which, not surprisingly, has been the subject of decades of research.

A simple way to proceed is to assume an analytical form of the interaction, such as a Lennard-Jones or Morse potential. However, extensive studies have shown that pair potentials fail to capture some of the main features of solids and their surfaces. A very practical alternative was introduced some years ago in the form of semiempirical many-body potentials, such as those from the embedded atom method (EAM) [3.21, 3.6] and effective medium theory (EMT) [3.22], which continue to be widely applied. Although based on fitting of parameters from experiments (and sometimes from more sophisticated theoretical methods) these interaction potentials have had tremendous success in predicting/explaining the characteristics of surfaces and nanostructures of six transition metals (Ni, Cu, Ag, Pd, Pt, Au). Of course, their reliability needs to be checked by comparison with more accurate methods, such as those based on ab-initio density functional theory. Below, we give some general details of ab-initio electronic structure calculations that we and others have used. We follow this up with a short description of the EAM potentials that have been used to calculate structural properties of vicinal surfaces.

4.1.1 Ab-initio Electronic Structure Calculations

Since this topic is covered in detail in several other chapters, only a summary of calculational details is provided here. Ab-initio electronic structure calculations based on density functional theory (DFT) are performed either within a pseudopotential or an all-electron approach using either the local-density approximation (LDA) [3.23] or the generalized gradient approximation (GGA) [3.24]. Of the variety of DFT based codes that are available, two that are frequently used here are the Vienna ab-initio simulation package (VASP) [3.25] and the Quantum ESPRESSO [3.26]. For example, to model the charge density distribution and structural relaxation of nanoparticles, Shafai et al [3.27] adopted the pseudopotential approach [3.25] to describe the interaction of the electrons and the nucleus employing projector-augmented wave (PAW) [3.28, 3.29] pseudopotentials. For systems in which relativistic effects are significant, as in gold, scalar relativistic pseudopotentials are used [3.30, 3.31, 3.32, 3.33]. The Kohn–Sham orbitals are expanded as plane waves with a reasonable kinetic energy cutoff (for example, 720 eV for Au). The exchange-correlation functional proposed by Perdew, Burke, and Ernzerhof (PBE) [3.34] was used. The ion cores in the system are allowed to relax using a conjugate gradient algorithm until the Hellman–Feynman forces acting on each atom are less than ${\mathrm{1\times 10^{-3}}}\,{\mathrm{eV/\AA}}$. Similarly, the total energy is taken to be converged if the changes in the energy in the self-consistent loop are below 10⁻⁶ eV. Since we use periodic boundary conditions, the supercell for the system of interest is taken to be large enough such that the interactions between the model system and its periodic images are negligible. That is in addition to a specific number of layers of atoms, the model system contains about 12 − 15 Å of vacuum. The resulting supercells, thus, have dimensions that depend on the size/shape of the system under study. Further details of these calculations may be found elsewhere [3.10].

4.1.2 Many-Body Interaction Potentials

To describe the interactions between the atoms in model systems, several types of many-body, semiempirical potentials [3.21, 3.22, 3.35, 3.36, 3.6] have been proposed. Without doing justice to others, some details of the semiempirical, many-body potential, EAM, developed by Foiles, Baskes, and Daw [3.21, 3.6] are presented. Although the EAM potentials neglect the large gradient in the charge density near the surface and use atomic (elemental) charge densities to describe those for the corresponding solid, for the six fcc metals Ag, Au, Cu, Ni, Pd, and Pt, and their alloys, these interatomic potentials have done a good job of reproducing many of the characteristics of bulk and surface systems [3.21, 3.6]. We have also found EAM potentials to be reliable for examining the temperature-dependent structure and dynamics of the low Miller index surfaces of Ag and Cu [3.37, 3.38, 3.39, 3.40, 3.41] and also for describing the energetics of vicinal surfaces of Cu and self-diffusion processes on the (100) surfaces of Ag, Cu, and Ni [3.42]. This method exploits the findings that the ground state of an interacting electron gas is a unique functional of the total electron charge density [3.23]. It further assumes that the energy of an impurity in a host is a functional of the electron density of the unperturbed host electron density [3.43].

The many-body term in the total energy has the form E = F_Z,R(ρ_h(r)), where ρ_h(r) is the electron density of the host without impurity at R, the position at which the impurity is to be replaced, and Z is the type of impurity. Each atom in a solid is, thus, viewed as an impurity embedded in a host consisting of all other atoms, and its energy is given by

$$E_{\mathrm{tot}}=\sum_{i}F_{i,j}(\rho_{\mathrm{h},i})+\frac{1}{2}\sum_{\substack{i,j\\ (i\neq j)}}\varnothing_{i.j}(R_{i,j})\;,$$

(3.15)

where ∅_i.j is the short-range pair potential ($\varnothing_{i.j}=Z_{i}(r)Z_{j}(r)/r$) and R_i,j is the distance between atoms i and j. Further simplification is introduced by assuming that the host density (ρ_h,i) is a sum of the atomic densities (ρ ^a_j ) and given by

$$\rho_{\mathrm{h},i}=\sum_{j(\neq i)}\rho_{j}^{\mathrm{a}}(R_{ij})\;,$$

(3.16)

where ρ ^a_j is the contribution to the electron density from atom j. Connection to the atomic densities makes the total energy to be a function of the atomic positions in electron density ρ_i, and ∅ is a short-range electrostatic pair potential between atoms i and j.

It is further assumed that the electron density ρ_i is given by a linear superposition of the electron densities of the constituent atoms, which is taken to be spherically symmetric. The EAM functions for the six fcc metals, Ag, Au, Cu, Ni, Pd, and Pt, and their alloys were developed by numerically fitting the functions to the bulk lattice constants, cohesive energy, elastic constants, vacancy formation energy, and alloy heats of mixing. Using these interactions for a model system constructed in its bulk terminated positions, the conjugate gradient method is used to relax the system to 0 K equilibrium configuration. The dynamical matrix needed to calculate the vibrational (or phonon) density of states for a system of interest is then obtained from analytical expressions for the partial second derivatives of the EAM potentials [3.21, 3.6].

4.2 Determination of the Vibrational Density of States

There are several theoretical techniques for calculations for surface and bulk phonons in crystals, ranging from those based on DFT to those that rely on some form of interatomic potentials. Ab-initio methods based on density functional perturbation theory and their applications are covered elsewhere in this Handbook and, hence, will not be covered here. Instead, we will focus here on methods that typically employ semiempirical interaction potentials. In these cases, the most commonly used is the slab method in which one needs to diagonalize the dynamical matrix portraying the interactions between the particles in N layers of the slab [3.44]. In the case of high Miller index surfaces, in particular those with large terraces, representative of large surface periodicities, slab calculations are hampered by the need for model systems with an extraordinarily large number of layers to describe the inhomogeneities of the surface structure. Furthermore, our interest is in calculating the phonon density of states, rather than phonon dispersion curves, of systems that lack long-range order, because of the presence of steps, kinks, etc. For such purposes, the continued fraction method [3.45] using a real space Green’s function developed by Wu and coworkers [3.46] is especially suitable, since it determines the local densities of states (LDOS) directly. In fact, the total phonon density of states is calculated as a sum of the contributions from the individual regions of interest. For example, the regions of interest could be the surface layer, the second layer, and so on. Or it could be the kink sites, the steps, the terraces, and so forth. The approach is based on constructing the resolvent matrix of an infinite block-tridiagonal matrix [3.45]. Green’s function is defined as the matrix representation of the resolvent operator that yields the LDOS via its matrix elements. Such a matrix must be constructed in a block-tridiagonal form, and the range of the interactions must be finite, which is fortunately the case for the transition metals named above. The system Hamiltonian is then written in a block-tridiagonal matrix as

$$\textbf{H}=\begin{pmatrix}\ddots&&&\\ \boldsymbol{v}_{i,i-1}&\textbf{h}_{i}&\boldsymbol{v}_{i,i+1}&\\ &\boldsymbol{v}_{i+1,i}&\textbf{h}_{i+1}&\boldsymbol{v}_{i+1,i+2}\\ &&&\ddots\\ \end{pmatrix}\;,$$

(3.17)

where h_i submatrices along the diagonal are 3n_i × 3n_i+1 square matrices, v_i,i+1 matrices along the off-diagonals have dimension 3n_i × 3n_i+1, and n_i is the number of particles in the chosen locality. Figure 3.6 shows such a system divided into three regions (localities labeled L₁ to L₃). For each locality, there is a submatrix (h₁, h₂ and h₃) describing the interactions within the associated locality. The submatrices (v₁₂ and v₂₃) describe the interactions between localities. The interactions beyond these localities are assumed to be bulk-like.

Fig. 3.6

Localities and submatrices. (Adapted with permission from H. Yildirim, Dissertation, UCF 2010)

Note that in our calculations, the system Hamiltonian is described by the force constant matrix, which is obtained from the second derivative of the interaction potential. The eigenvalue can be derived by means of the Green’s function that is associated with the matrix H and is given by

$$\textbf{G}(z)=(z\textbf{I}-\textbf{H})^{-1}\;,$$

(3.18)

where z = ω² + iε, ε is the width of the Lorentzian representing the delta function at ω², and I is a unit matrix of the same dimension as that of H. The diagonal element of the Green’s function matrix corresponding to a chosen locality is expressed as

$$\begin{aligned}\displaystyle\textbf{G}_{ii}&\displaystyle=[(z\textbf{I}-\textbf{h}_{i})-\boldsymbol{v}_{i,i+1}\Delta_{i+1}^{+}\boldsymbol{v}_{i+1,i}\\ \displaystyle&\displaystyle\qquad-\boldsymbol{v}_{i,i-1}\Delta_{i-1}^{-}\boldsymbol{v}_{i-1,i}]^{-1}\;.\end{aligned}$$

(3.19)

Δ ⁺_i and Δ ⁻_i are defined as forward and backward partial Green’s functions and described by

$$\begin{aligned}\Delta_{i}^{+} & =(z\textbf{I}-\textbf{h}_{i}-\boldsymbol{v}_{i,i+1}\Delta_{i+1}^{+}\boldsymbol{v}_{i+1,i})^{-1}\;,\end{aligned}$$

(3.20)

$$\begin{aligned}\Delta_{i}^{-} & =(z\textbf{I}-\textbf{h}_{i}-\boldsymbol{v}_{i,i-1}\Delta_{i-1}^{-}\boldsymbol{v}_{i-1,i})^{-1}\;.\end{aligned}$$

(3.21)

The relation between the successive diagonal elements of the Green’s function matrix G is obtained by

$$\textbf{G}_{ii}=\Delta_{i}^{\pm}+\Delta_{i}^{\pm}\boldsymbol{v}_{i,i\pm 1}\textbf{G}_{i\pm 1,i\pm 1}\boldsymbol{v}_{i\pm 1,i}\Delta_{i}^{\pm}\;.$$

(3.22)

As can be seen the calculation of Green’s function mainly depends on the forward and backward Green’s functions (Δ ^±_i ), which are the inverses of matrices with the dimensions same as that of h_i. The convergence procedure for the calculation of Δ ^±_i for an infinite system starts with the condition as

$$\Delta_{1}^{+(1)}=\lim_{\substack{m\rightarrow\infty,\\ \varepsilon\rightarrow 0}}\Delta_{1}^{+(m)}(z)\;,$$

(3.23)

where

$$\begin{aligned}\displaystyle\Delta_{1}^{+(m)}&\displaystyle=\{(zI_{1}-h_{1})-v_{12}[(zI_{2}-h_{2})-\ldots{}[v_{m-1,m}\\ \displaystyle&\displaystyle\qquad\times(zI_{m}-h_{m})^{-1}v_{m,m-1}]^{-1}\ldots]^{-1}v_{21}\}^{-1}\;.\end{aligned}$$

(3.24)

From this equation, one can define the forward Green’s function for any chosen regime as

$$\begin{aligned} & \Delta_{1}^{+(1)}=(zI_{1}-h_{1})^{-1}\;,\end{aligned}$$

(3.25)

$$\begin{aligned} & \Delta_{1}^{+(2)}=[(zI_{1}-h_{1})-v_{12}(zI_{2}-h_{2})^{-1}v_{21}]^{-1}\;,\end{aligned}$$

(3.26)

$$\begin{aligned} & \begin{aligned}\displaystyle&\displaystyle\Delta_{1}^{+(3)}=\{zI_{1}-h_{1}-v_{12}[(zI_{2}-h_{2})\\ \displaystyle&\displaystyle\qquad\qquad-v_{23}(zI_{3}-h_{3})^{-1}v_{32}]^{-1}v_{21}\}^{-1}\;,\end{aligned}\end{aligned}$$

(3.27)

$$\begin{aligned} & \begin{aligned}\displaystyle&\displaystyle\;\,\vdots\\ \displaystyle&\displaystyle\Delta_{1}^{+(m)}\end{aligned}\;.\end{aligned}$$

(3.28)

The convergence procedure shows that the information obtained in previous steps cannot be used to calculate Δ ^+(m)₁ for the current step; hence, for each step, the sequence must be repeated independently. This requires excessive computing time. In order to make use of the information obtained at the previous step (Δ ^+(m−1)₁ ) and the current one (Δ ^+(m)₁ ), a recursive method is introduced that defines the forward Green’s function for the localities (Δ ⁺⁽¹⁾₁ , Δ ⁺⁽²⁾₁ , Δ ⁺⁽³⁾₁ , …, Δ ^+(m)₁ ) for a finite system as

$$\begin{aligned}\displaystyle\Delta_{1}^{+(1)}&\displaystyle=(zI_{1}-h_{1})^{-1}\equiv\Delta_{1}^{-}\;,\\ \displaystyle\Delta_{1}^{+(2)}&\displaystyle\equiv G_{11}^{(2)}\;,\\ \displaystyle\Delta_{1}^{+(3)}&\displaystyle\equiv G_{11}^{(3)}\;,\\ \displaystyle&\displaystyle\;\;\vdots\;,\\ \displaystyle&\displaystyle\;\;\vdots\;,\\ \displaystyle\Delta_{1}^{+(m)}&\displaystyle\equiv G_{11}^{(m)}\;,\end{aligned}$$

(3.29)

where G ^(m)₁₁ is the (1,1) diagonal block of Green’s function corresponding to the matrix H^(m). Using (3.22), Δ ⁺⁽²⁾₁ and G ⁽²⁾₂₂ are given by

$$\begin{aligned}\Delta_{1}^{+(2)} & =G_{11}^{(2)}=\Delta_{1}^{-}+\Delta_{1}^{-}v_{12}G_{22}^{(2)}v_{21}\Delta_{1}^{-}\;,\end{aligned}$$

(3.30)

$$\begin{aligned}G_{22}^{(2)} & =\Delta_{2}^{-}=[(zI_{2}-h_{2})-v_{12}\Delta_{1}^{-}v_{21}]^{-1}\;.\end{aligned}$$

(3.31)

So, the relation between the successive forward Green’s functions is obtained as

$$\Delta_{1}^{+(2)}=\Delta_{1}^{+(1)}+A_{1}\Delta_{2}^{-}B_{1}\;,$$

(3.32)

where A₁ = Δ ⁻₁ v₁₂ and B₁ = v₁₂Δ ⁻₁ . Repeating the same steps, general recursive relation can be obtained

$$\begin{aligned}\Delta_{1}^{+(m)} & =\Delta_{1}^{+(m-1)}+A_{m-1}\Delta_{m}^{-}B_{m-1}\;,\end{aligned}$$

(3.33)

$$\begin{aligned}A_{m-1} & =A_{m-2}\Delta_{m-1}^{-}v_{m-1,m}\;,\end{aligned}$$

(3.34)

$$\begin{aligned}B_{m-1} & =v_{m,m-1}\Delta_{m-1}^{-}B_{m-2}\;.\end{aligned}$$

(3.35)

This method, thus, simplifies the calculation of Green’s functions to inversion and multiplication of matrices whose dimensions are much smaller than the total number of degrees of freedom of the system. The diagonal element of Green’s function represents the entire system. The method ensures that Green’s function associated with a particular locality in the system can be reasonably calculated. Further details may be found in [3.46, 3.47, 3.48, 3.49].

In calculating the LDOS for the surfaces of interest, we take each system to consist of an infinite number of layers with in-plane periodicity and specify a certain number of layers to constitute a locality. The submatrix elements of the block-tridiagonal matrix represent the force constants between the atoms, within and between the chosen localities. We then determine the normalized LDOS from the trace of Green’s function by employing

$$\begin{aligned}g(\nu^{2}) & =\frac{-1}{3n\uppi}\lim_{\varepsilon\rightarrow 0}\mathrm{Im}\{\mathrm{tr}[G(\nu^{2}+\mathrm{i}\varepsilon)]\}\;,\end{aligned}$$

(3.36)

$$\begin{aligned}N(\nu) & =2\nu g(\nu^{2})\;,\end{aligned}$$

(3.37)

where ε is the width of the Gaussian and frequency-dependent LDOS (N(ν)) is related to the normalized local LDOS (g(ν²)).

The method has been applied successfully to both high and low-symmetry systems [3.47, 3.48, 3.49, 3.50, 3.51, 3.52] such as bulk, low and high Miller index surfaces, and single-element [3.53, 3.54] and bimetallic NPs [3.55]. These studies have proved that it can accurately reveal the vibrational properties, especially for low-symmetric systems. Applications of the method have provided insights into the effect of coordination and alloying on vibrational and thermodynamical properties of such low-symmetry systems [3.49].

4.3 Determination of Surface Thermodynamics

Having calculated the vibrational density of states, as described above, the task is to determine the surface thermodynamical functions, which includes vibrational entropy. Since the harmonic approximation of lattice dynamics was invoked in the above section, in what follows, the same approximation will be implicit. We expect the results summarized here to be valid for low temperatures. Molecular dynamics simulations for Cu and Ag surfaces [3.37, 3.39, 3.40], using EAM potentials, show the atomic mean-square displacements to vary linearly with temperature even beyond half the melting temperature, implying the validity of the harmonic approximation for that temperature range. Below, we provide some details of the calculations of the local and excess vibrational free energy for surfaces. The main quantity of interest here is the vibrational contribution to surface free energy. While details may be found in [3.17], we include the main points here.

As discussed before, the free energy of a system is given by the standard definition F = U − TS, where U is the internal energy, S is the entropy, and T is the temperature. Both U and S have contributions from atomic configurations and vibrations, such that F = F_conf + F_vib. That is, for each atomic configuration of the system, there is a specific vibrational contribution, which can be further written as F_vib = U_vib − TS_vib. It is this latter quantity that we calculate here for the surface systems of interest for their relaxed atomic configurations (of minimum energy) at 0 K. In the harmonic approximation, the vibrational free energy, the vibrational internal energy, and the vibrational entropy are given by

$$\begin{aligned}F^{\mathrm{vib}} & =k_{\mathrm{B}}T\int_{0}^{\infty}N(\nu)\ln\left[2\sin\mathrm{h}\left(\frac{x}{2}\right)\right]\mathrm{d}\nu\;,\end{aligned}$$

(3.38)

$$\begin{aligned}U_{\mathrm{vib}} & =k_{\mathrm{B}}T\int_{0}^{\infty}N(\nu)\left(\frac{1}{2}x+\frac{x}{\mathrm{e}^{x}-1}\right)\mathrm{d}\nu\;,\end{aligned}$$

(3.39)

$$\begin{aligned}S_{\mathrm{vib}} & =k_{\mathrm{B}}\int_{0}^{\infty}N(\nu)\left[-\ln(1-\mathrm{e}^{-x})+\frac{x}{\mathrm{e}^{x}-1}\right]\mathrm{d}\nu\;,\end{aligned}$$

(3.40)

where k_B is the Boltzmann constant, T is the temperature, x = hν ∕ (k_BT), and h is the Planck constant. The vibrational density of states can, in turn, be written as N(ν) = ∑_ln_l(v), where n_l(v) is the local density of states of the region of interest, the layer, the step, the kink, or any other locality.

To calculate the vibrational contribution to the free energy of a surface (flat, stepped, kinked, …) we consider a slab of N atoms, arranged such that the top and bottom surfaces consist of the Miller index planes. The surface free energy, which is defined as the excess over the values associated with the bulk system, is evaluated using the expression

$$F_{\mathrm{SURF}}^{\mathrm{vib}}=\frac{F_{\mathrm{SLAB}}^{\mathrm{vib}}-NF_{\mathrm{BULK}}^{\mathrm{vib}}}{2A}\;.$$

(3.41)

In (3.41), F ^vib_SLAB and F ^vib_BULK are, respectively, the vibrational free energy calculated for a finite slab representing the surface system and a single bulk atom, and A is the surface area of the system. The factor 2 arises from the fact that the system contains two free surfaces. In the same vein, the step vibrational free energy is defined as the excess energy over that for a low Miller index surface of the same geometry as the terrace. The vibrational contribution to the step free energy is then calculated using [3.3]

$$F_{\mathrm{STEP}}^{\mathrm{vib}}=F_{\mathrm{S}}^{\mathrm{vib}}(\theta)-(p-1+f)F_{\mathrm{S}}^{\mathrm{vib}}(\infty)\;,$$

(3.42)

where F ^vib_STEP is the vibrational step free energy, F ^vib_S (θ) and F ^vib_S (∞) are the vibrational surface energies of the corresponding high and low Miller index surfaces forming the terraces of the vicinal surface, p is the number of atomic chains on the terrace, and f is a geometrical factor determined by the projection of the ledge on the terrace. According to this definition f is 2 ∕ 3 for (211), 1 ∕ 3 for (331), and 1 ∕ 2 for (511).

5 Results

In this section, we present examples of vibrational density of states for some prototype low and high Miller index surfaces. For the former, we consider the (100) surfaces of Pd, Pt, and Au, for which an extensive study was carried out in [3.10] to compare the results obtained from application of the EAM potentials with those from DFT, while for the latter we focus on Ni(977).

5.1 Local Vibrational Density of States

The layer-resolved vibrational density of states for the first four layers of Pd(100), Pt(100), and Au(100) are presented in Fig. 3.7a-c. We note from the figure that the LDOS for the atoms in the first layer is remarkably different from that for the atoms in the second, third, and fourth layers. It, thus, suffices to focus our analysis of the LDOS only for atoms in the first layer. The reader is reminded that the first-layer atoms on the fcc(100) surface have coordination 8, since they have lost four neighbors. As a result, there is a loosening of bonds and a softening of the in-plane force constants in the first layer. The softening of the force constants, in turn, yields a shift towards lower frequencies that is reflected in Fig. 3.7a-c. Note that this red shift in the density of states is a characteristic of all the five elements discussed in this chapter. This is not a global red shift of the density of states, which would have had drastic consequences for the stability of the surfaces. Rather, it is depletion of the high-frequency band accompanying an enhancement of the low-frequency band. We can also conclude from Fig. 3.7a-c that these effects are marginal for the second-layer atoms and are absent for the third and fourth layers. Though not presented in Fig. 3.7a-c, the LDOS for Cu(100) and Ag(100) show the same trend as the three surfaces shown therein.

Fig. 3.7a-c

Layer-resolved vibrational densities of states for (a) Pd(100), (b) Pt(100), and (c) Au(100). (Reprinted from [3.10], with permission from Elsevier)

Turning next to high Miller index surfaces, we present the calculated local density of states for the step and the terrace atoms of Ni(977) and compare them with those for the Ni(111) surface atoms in Fig. 3.8. We have chosen here Ni(977), since this surface displays a step localized vibrational mode at a frequency [3.52] that is close to that of one that was observed experimentally using the He-atom surface scattering method [3.56, 3.57]. Note that the terrace of Ni(977) contains eight atoms, seven of which have the same coordination (9) as that of the atoms in the top layer of Ni(111), and the eighth is at the step with coordination 7. Because of the reasonably large terrace width, the LDOS of the Ni(977) terrace atoms is similar to that of the atoms in the top layer of Ni(111). This result suggests that in studies of phonons of vicinal surfaces with relatively large terraces, as in (977), or on realistic surfaces with nonregularly distributed steps, it is feasible to use a local approach to determine the phonons of the step and its surrounding atoms, while resorting to standard techniques (diagonalizing a force constant-based dynamical matrix for a slab with a large number of layers) for calculating phonons of the close-packed flat surface for the terrace. Another interesting feature in Fig. 3.8 is that the low-frequency modes associated with the step atoms display a global shift to lower frequencies, to the modes associated with the terrace (and Ni(111) surface atoms), thereby pointing to an extra softening of relevant force constants, in agreement with suggestions of Niu et al [3.56, 3.57]. As shown in Table 3.2, the force constants between the surface atoms, calculated using EAM interaction potentials, undergo both softening and stiffening.

Fig. 3.8

Projected vibrational densities of states for the Ni(977) step, terrace, and Ni(111) surface atoms. (Reprinted from [3.52], with permission of AIP Publishing)

Table 3.2 Force constant matrices (in eV Å⁻² ∕ unit mass) between the step atom and it neighbors on Ni(977). (Adapted from [3.17], © 2003 IOP Publishing)

A detailed examination of the LDOS of Ni(977) [3.52] shows that the most prominent low-frequency mode is at 3.3 THz with maximum displacement along the x-direction. The displacement vectors of the atoms contributing to this mode, found from the imaginary part of the Green function, is shown in Fig. 3.9. Note that this mode is quasi-one dimensional: it involves only the concerted motion of the SC and CC atoms in conjunction with that of the TC atoms. The step atoms alternate with displacement vectors $(+1,0,-0.67)$ and $(-1,0,+0.67)$, the corner atoms with vectors $(+0.27,0,-1)$ and $(-0.27,0,+1)$, and the TC atoms with vectors (0 ,  + 0.24,0) and (0 ,  − 0.24,0). Thus, the step and the corner atoms move in the xz-plane, while the TC atoms move along the y-direction to accommodate the propagation of this mode along the direction parallel to the step. The rest of the atoms are at rest. The wavelength of the mode is 2 ×  nearest-neighbor distance or 4.978 Å, and its wavevector $q=(0.0,{\mathrm{1.262}}\,{\mathrm{\AA{}^{-1}}},0.0)$. This is a superlocalized mode, since the motion is restricted to the step atoms and their immediate neighbors on the surface. Modes such as the one displayed in Fig. 3.9 exist on a number of vicinal surfaces. Similarly, along the y and z-directions there are noticeable peaks, at low frequencies, at 3.9, and 4.9 THz, respectively. There is also a prominent high-frequency peak at 9.1 THz, close to the top of the bulk band. In Table 3.3, we present a summary of the frequencies of the observed modes and refer the reader to discussions in related publications.

Fig. 3.9

The displacement pattern of the step-localized vibrational mode on Ni(977). (Adapted from [3.18], © IOP Publishing. Reproduced with permission. All rights reserved)

Table 3.3 Observed (Exp.) and calculated (Th.) vibrational modes on metal vicinal surfaces, in THz. (Adapted from [3.17], © IOP Publishing)

5.2 Surface Vibrational Free Energies

An important conclusion from the calculated LDOS of the (100) surfaces of Pd, Pt, and Au presented in Fig. 3.7a-c is that we should expect deviations in surface thermodynamics from the bulk values, and that this difference will be mainly for the top-layer atoms. This inference (obtained using DFT) is in accord with what was reported in earlier publications on the vibrational dynamics and thermodynamics of vicinal and kinked fcc metal surfaces using EAM potentials [3.21, 3.6]. The quantities of interest here are the lattice heat capacity (local and excess), the contribution of the vibrational dynamics to the free energy (local and excess), and the atomic mean-square displacements. As shown in Figs. 3.10a-ea, 3.11a-ea and 3.12a-ea, the local lattice heat capacity (C_V) of the first-layer atoms of Pd(100), Pt(100), and Au(100) differs from that of the other atoms in the system, and that this difference is temperature dependent. These deviations from the bulk values are better described by the local excess from the bulk as illustrated in Figs. 3.10a-eb, 3.11a-eb, and 3.12a-eb. Indeed, for Pd and Pt(100), the maximum deviation was found to be 3.4 and 2.8 J ∕ (K mol), respectively, both occurring at a temperature of 50 K. However, the first-layer atoms of Au(100) behave differently; with a maximum deviation of only 2.2 J ∕ (K mol) occurring at a lower temperature (30 K).

Fig. 3.10a-e

The thermodynamic functions for Pd(100): (a) lattice heat capacity; (b) excess lattice heat capacity; (c) vibrational contribution to the free energy; (d) excess vibrational contribution to the free energy; (e) vibrational mean-square amplitude. (Reprinted from [3.10], with permission from Elsevier)

Fig. 3.11a-e

The thermodynamic functions for Pt(100): (a) lattice heat capacity; (b) excess lattice heat capacity; (c) vibrational contribution to the free energy; (d) excess vibrational contribution to the free energy; (e) vibrational mean-square amplitude. (Reprinted from [3.10], with permission from Elsevier)

Fig. 3.12a-e

The thermodynamic functions for Au(100): (a) lattice heat capacity; (b) excess lattice heat capacity; (c) vibrational contribution to the free energy; (d) excess vibrational contribution to the free energy; (e) vibrational mean-square amplitude. (Reprinted from [3.10], with permission from Elsevier)

Let us now turn our attention to the local and excess vibrational free energy. The results are presented in Figs. 3.10a-e, 3.11a-e, and 3.12a-e, for Pd, Pt, and Au(100), respectively. Here again, only the first-layer atoms show differences from the atoms in the other layers. As shown in the figures, the local contributions to the vibrational free energies for the layers beyond the second converge to the corresponding bulk values. For the atoms in the first layer, the local contribution to the vibrational free energy decreases with temperature and reaches −48, −64, and −79 meV ∕ atom for Pd, Pt, and Au(100), respectively, at 300 K. The excess vibrational free energies are significantly different for the first-layer atoms, as can be seen in Figs. 3.10a-ed, 3.11a-ed, and 3.12a-ed, with the contribution amounting to 19, 17, and 14 meV ∕ atom at 300 K. Our DFT calculations for Cu and Ag(100) show 15 meV ∕ atom, which is in good agreement with the EAM results (about 18 meV ∕ atom for both surfaces at 300 K).

In the harmonic approximation, the mean-square vibrational amplitude (MSVA) is expected to vary linearly with temperature. Note that at 0 K, the MSVA does not go to zero due to the zero-point motion. In Figs. 3.10a-ee, 3.11a-ee, and 3.12a-ee, we present our DFT results for the MSVA of Pd, Pt, and Au(100), respectively. The MSVA of the atoms in the third and fourth layers are bulk-like (0.008, 0.0075, and 0.0130 Ų at 300 K for Pd, Pt, and Au(100), respectively). Though MSVA of the second-layer atoms is close to the bulk values, that corresponding to the first-layer atoms shows large deviations. The ratio between the first-layer and the bulk MSVA is 2.06, 1.73, and 1.73 for Pd, Pt, and Au(100), respectively. This deviation is due to the decrease of coordination at the surface as compared to the bulk. Note that the ratio is the same for Pt and Au(100) and the highest for Pd(100), reflecting the mass effect.

We now turn to a summary of the impact of further undercoordination on the vibrational contribution to the thermodynamic properties as presented by vicinal surfaces with a few specific examples. In Fig. 3.13a-c, we show the vibrational LDOS for the step (SC), terrace (TC), and corner (CC) atoms of Cu(511), Cu(211), and Cu(331). The shift towards low frequencies, in each case, reflects the lower coordination of these atoms. There are some differences in LDOS between the sets of atoms on the three surfaces. On Cu(331), for example, the corner atom is almost bulk-like, while the other two atoms contribute more to the lower-frequency modes than the atoms in the bulk. As noted in previous publications [3.11, 3.3], the LDOS of the (211) and the (511) surfaces are similar to each other and different from those of the (331). These similarities and differences are related to trends in the multilayer relaxation patterns on these surfaces, as discussed above. Furthermore, the plots in Fig. 3.13a-c suggest complexity in the way the different surface localities differ from those of the atoms in the bulk. We also find that the LDOS of the BNN atoms (in the layer adjacent to the corner atoms) show an enhancement in the high-frequency region over bulk values [3.3]. This enhancement at the high-frequency end is very prominent in the LDOS of the surface atoms on Cu(532), which has a regular array of kinks, as can be seen in Fig. 3.14.

Fig. 3.13a-c

Projected vibrational densities of states of the step (SC), terrace (TC), and corner (CC) atoms on (a) Cu(511), (b) Cu(211), and (c) Cu(311) compared to that of a bulk atom. (Adapted from [3.17], © IOP Publishing. Reproduced with permission. All rights reserved)

Fig. 3.14

The projected vibrational density of states of kink sites on Cu(532). (Adapted from [3.17], © IOP Publishing. Reproduced with permission. All rights reserved)

The effect of new features in the vibrational dynamics is reflected in the calculated local vibrational free energy illustrated in Fig. 3.15a-c for the top layers of (511), (211), and (331). On Cu(511) and Cu(211), the local contribution to vibrational entropy from atoms in the three surface chains (SC, TC, CC) is distinct from that of the atoms in the bulk. On Cu(331), however, layer 3, which corresponds to CC atoms on the terrace, shows characteristics more akin to those of atoms in the bulk than those on the surface. This interesting behavior can, again, be traced to the effective coordination number. As we know from Table 3.2, the coordination number for a CC atom of (511) and (211) is 10, while that of a (331) surface is 11, close to the value 12 for a bulk atom. In the equilibrium configuration obtained after ionic relaxation of the model system [3.11, 3.3], while the CC atom of (211) and (511) moves upward, away from the bulk, that of (331) moves toward the rest of the bulk, increasing its effective coordination number. Note that the difference in coordination between the terrace atoms on (511) and (211) (8 and 9, respectively) is also displayed in the local free energies in Fig. 3.15a-c. The terrace atoms on (511) have contributions closer to those of their step atoms, while on (211), they are closer to those of the corner atoms. The reader is referred to [3.3] for details of the effects of varying coordination and complex relaxation patterns on the vibrational thermodynamic properties of the vicinal surfaces. We summarize the essential points below.

Fig. 3.15a-c

The dependence of the local contribution on the vibrational free energy F_vib of vicinal surfaces on the local coordination for (a) Cu(511), (b) Cu(211), and (c) Cu(331). (Adapted from [3.17], © IOP Publishing. Reproduced with permission. All rights reserved)

For convenience, we focus the discussion on local thermodynamical quantities calculated at 300 K. Interestingly, the contributions of the step atoms (SC) are about the same on all three vicinal surfaces. On the Cu surfaces, it is almost 41 meV ∕ atom, while on the Ag surfaces, it is about 68 meV ∕ atom. Although the percentage enhancement over the bulk value is larger on Cu surfaces than on Ag, in each case, it is in excess of about 19 meV ∕ atom over that in the bulk. Turning to the contributions of the TC atoms, we find a good correspondence with their respective coordination number. On (511), on which their coordination is 8, they contribute about 17 meV ∕ atom more than the bulk atoms, similarly to the value for the atoms on (100) surface with the same coordination. On (211) and (331), their contribution over the bulk is about 13 − 11 meV ∕ atom, resembling that of the surface atoms on (111), which also have a coordination of 9. The largest difference in contribution for the atoms on the three types of vicinal surface comes from the corner atoms, which contribute only 2 − 3 meV ∕ atom more than the atoms in the bulk on Cu(331), while on Cu(511) and Cu(211) the corresponding amount is 9 meV ∕ atom. This is, again, understandable from previous arguments based on atomic coordination. The atoms in the other layers in the tables have similar contributions to the bulk, and small variations can, again, be linked to their ionic relaxation and nonbulk-like contribution to the vibrational DOS. The above findings regarding the differential, local vibrational free energy contribution of atoms in undercoordinated sites show that the vibrational free energy does not scale with the coordination number in a simple manner, although its dependence on atomic coordination is remarkable. This is not surprising, as other factors also impact the characteristics of the surface atoms. Ionic relaxations of the individual atoms and their bond lengths with the surrounding atoms are also expected to play a role in defining the local characteristics of a surface. While atoms on low Miller index surfaces lose symmetry in the direction perpendicular to the surface, the atoms forming the terraces of vicinal surfaces experience a lack of symmetry along the direction perpendicular to the step edge, in the terrace plane, in addition to the direction normal to it. This, in turn, leads to complex relaxation patterns and bond lengths for the surface atoms, as shown in previous studies [3.11, 3.22], resulting in distinctive characteristics of the local regions on the surface. Note that in this chapter, relaxation patterns of the surfaces considered are not presented, to keep the focus on vibrational contributions. However, there is a direct link between frequencies of surface (and step) localized modes, the changes in force constants from the values in the bulk, and the manner in which atoms in undercoordinated sites relax to configurations away from their bulk terminated positions. The reader is encouraged to refer to previous studies that summarize these interesting and complex relaxation patterns.

Using the local contributions to the vibrational entropy we can calculate the surface and step excess free energies, following the procedure discussed earlier; we find the calculated surface excess vibrational free energies for Cu and Ag vicinals at 0, 100, and 300 K to be quite small. The calculation of the excess step free energies, however, shows a remarkable effect of the vibrational component. The temperature variation of the step vibrational free energy for the three surface geometries is plotted in Fig. 3.16a,b. It is remarkable that the excess vibrational contribution is almost nonexistent for both Ag(511) and Cu(511). From 0 to 300 K, it shows a variation of only about 1 meV ∕ atom. On the other hand, the contribution is remarkable for both Ag(211) and Cu(211) and constitutes about 10% of the total free energy [3.3, 3.58] in the case of Cu. The case of the (331) surfaces lies in between the other two geometries. Since the step free energy is relevant to the roughening temperature of a surface, we expect vibrational contributions to play an important role, particularly for Cu(211) and Ag(211).

Fig. 3.16a,b

The dependence of the excess step free energy (vibrational part) on the surface geometry. (Adapted from [3.17], © IOP Publishing. Reproduced with permission. All rights reserved)

6 Summary

This chapter contains some details of the calculation of local vibrational contributions of surface atoms to selected dynamical properties and thermodynamic quantities. Already published results obtained by the author and her coworkers (A. Kara, S. Durkanoglu, M. Alcantara Ortigoza, and H. Yildirim) are used as examples of the novel vibrational characteristics that impact the thermodynamical properties of a set of low and high Miller index surfaces of several transition metals. Calculations were performed using either ab-initio electronic structure calculations or many-body interaction potentials. Comparisons of results are made with experimental data where available. The main conclusion to be drawn is that of the important role of the local coordination and geometry in determining the structural, dynamical, and, ultimately, thermodynamical properties of these surface nanostructures. Of the large list of vicinals of varying surface geometry, terrace width, step-face orientation, and elemental metal summarized here, the striking feature in the multilayer relaxation is the large outward relaxation of the corner atom (actually, the least undercoordinated atom) coupled with the inward relaxation of its neighboring surface atom. At least for Cu, Pd, Al, and selected vicinals of Ni, Ag, and Pt, the trends in the calculated multilayer relaxations point to this enhanced local effect at the step and near the corner site, in accordance with principles of charge smoothing. The reduced coordination of the surface atoms leads to electronic densities of states that are narrower and shifted towards the Fermi level. Furthermore, changes in the bond lengths of the surface atoms from their neighbors are reflected in corresponding softening and stiffening of force constants with respect to those in the bulk. As a result, localized modes appear on stepped surfaces whose existence has been verified experimentally. Moreover, projected vibrational densities of states of surface atoms show enhancements of modes at the low-frequency end, as well as near (and sometimes above) the top of the bulk modes. The vibrational entropies of these surfaces are thereby impacted on by their local geometry and coordination. The local contributions scale with coordination in a complex manner. While the surface free energies of a set of vicinals of Ag and Cu show very little sensitivity to the vibrational contribution, the excess step free energy could have a vibrational contribution of about 25%.

In considerations of growth and surface stability, vibrational contributions cannot be ignored. Needless to say, surface relaxations are an inherent component of the novel features of vicinal surfaces and should not be ignored in any serious examination of these surfaces. Last but not least, the modifications in the structural, dynamical, and subsequently thermodynamical, properties introduced by the presence of steps on surfaces are local in nature, thereby allowing an extension of the knowledge gained from systematic studies of stepped surfaces to more complex environments such as those on nanocrystals.