Universität Ulm Abteilung Elektrochemie

Simulation of Electrochemical Nanostructures

Dissertation zur Erlangung des Doktorgrades Dr. rer. nat. der Fakultät für Naturwissenschaften der Universität Ulm

vorgelegt von Marcelo M. Mariscal aus Cordoba (Argentinien) 2004

Ulm, July 2004

Dekan: Prof. Dr. R.J. Behm

1. Gutachter: Prof. Dr. W. Schmickler 2. Gutachter: Prof. Dr. E. Spohr

to Mariela

Contents I

1

1 Introduction

3

2 Prerequisites from Solid State Physics

7

2.1

Crystal Structure . . . . . . . . . . . . . . . . . . . . . . . . . .

7

2.1.1

Surface structure of fcc metals . . . . . . . . . . . . . . .

8

2.1.2

Crystal growth . . . . . . . . . . . . . . . . . . . . . . .

9

II

13

3 Simulation Techniques

15

3.1

Molecular Dynamics . . . . . . . . . . . . . . . . . . . . . . . . 15

3.2

Monte Carlo . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18

3.3

3.2.1

Importance Sampling . . . . . . . . . . . . . . . . . . . . 19

3.2.2

The Metropolis algorithm . . . . . . . . . . . . . . . . . 20

3.2.3

Grand-Canonical ensemble . . . . . . . . . . . . . . . . . 23

Interatomic Potentials . . . . . . . . . . . . . . . . . . . . . . . . 25 3.3.1

Pair potentials . . . . . . . . . . . . . . . . . . . . . . . . 26

3.3.2

Many body potentials . . . . . . . . . . . . . . . . . . . . 27

6

CONTENTS

III

33

4 Nanodecoration of surfaces with a STM

35

4.1

Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

35

4.2

The Model and calculation details . . . . . . . . . . . . . . . . .

38

4.2.1

The system Pd on Au(111) . . . . . . . . . . . . . . . . .

41

4.2.2

The system Pb on Au(111) . . . . . . . . . . . . . . . . .

48

4.2.3

The systems Cu on Pt(111) and Ag on Pt(111) . . . . . .

55

4.2.4

The system Cu on Cu-island on Ag(111) . . . . . . . . .

59

4.2.5

Review of systems studied . . . . . . . . . . . . . . . . .

61

A few conclusion on cluster generation with a STM tip . . . . . .

63

4.3

5 Consequences of Tip-Surface interactions

69

5.1

Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

69

5.2

Calculation Method . . . . . . . . . . . . . . . . . . . . . . . . .

70

5.3

Influence of tip structure in homoatomic systems . . . . . . . . .

71

5.3.1

Tip-sample approach . . . . . . . . . . . . . . . . . . . .

72

5.3.2

Tip-Sample contact . . . . . . . . . . . . . . . . . . . . .

73

5.3.3

Retraction . . . . . . . . . . . . . . . . . . . . . . . . . .

75

5.4 5.5

Comparison with the behaviour of the forces in heteroatomic systems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

78

A few conclusions on STM characterization . . . . . . . . . . . .

80

6 Generation of Metal Nanowires

83

6.1

Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

83

6.2

Mechanical Method . . . . . . . . . . . . . . . . . . . . . . . . .

84

6.3

Electrochemical Method . . . . . . . . . . . . . . . . . . . . . .

88

6.3.1

Introduction - Experimental Background . . . . . . . . .

88

6.3.2

Atomistic Simulation - grand canonical Monte Carlo . . .

91

CONTENTS

7

IV

103

7 A Molecular Dynamic study of metal deposition

105

7.1

Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 105

7.2

The Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 108

7.3

Simulation results . . . . . . . . . . . . . . . . . . . . . . . . . . 113

7.4

7.3.1

Pt on Au(111) . . . . . . . . . . . . . . . . . . . . . . . . 115

7.3.2

Ag on Au(111) . . . . . . . . . . . . . . . . . . . . . . . 119

A few conclusions on metal deposition . . . . . . . . . . . . . . . 121

V

123

8 General conclusions

125

9 Zusammenfassung

129

VI Appendix

133

A

135

Bibliography

137

Acknowledgments

143

Curriculum Vita

145

List of Publications

148

8

CONTENTS

Part I

Chapter 1 Introduction In last half century, simulations have become a supplement to theoretical and experimental approaches in solid state, materials sciences and many other fields. In principle because the development of new models and perhaps the most important factor: the tremendous growth of the computer power. Basically simulations consist of a numerical solution of a given equation(s), followed by determination of structural, thermodynamic, and kinetic data through statistical treatment of the results, providing detailed and essentially exact information about the system, enabling one to go directly from a microscopic Hamiltonian to the macroscopic properties measured in experiments. Computer simulations can be used as a purely exploratory tool. This sounds strange. One would be inclined to say that one cannot "discover" anything by simulation because you can never get out what you have not put in. Computer discoveries, in this respect, are not unlike mathematical discoveries. In fact, before computers were actually available this kind of numerical charting of unknown territory was never considered [1].

Actually several experimental and theoretical research groups focus on minia-

4

C HAPTER 1. I NTRODUCTION

turization, perhaps because a fashion or a tend, in which the keyword is "nano" that is the same to say "NanoScience" or "Nanotechnology" depends on the focus that the problem is taken. Conventional and recently new experimental techniques such probe microscopies are applied in order to fabricate new devices or "nano-devices" composed by a very small amount of atoms or molecules. Within this kind of nano-devices we can find at the present: nanoclusters, nanowires, nanogaps, nanodots, nano-gears, and many other small things that we can imagine.

At the present, the main challenge of scientists is to fabricate this kind of new devices following a general study about the applicability in future technologies. The application of these nanodevices depends at first view, on the thermodynamical and kinetic stability, and secondly on the costs to produce these nanodevices in large scale. Surprisingly, many of these nanodevices show a serie of phenomena never observed before, and one could speculate that the atomic size of these new devices play an important role, and make this field very interesting to an atomic detailed level computer simulation study.

In (1997) D. Kolb and coworkers [25] presented a new technique to produce small metal "nano"clusters on an electrode surface with the aid of a Scanning Tunneling Microscope (STM) in an electrochemical environment. The basis of the method was not clear at the early stages, and the unusual stability of the generated cluster was a great controversy between several research groups. The aim of a part of this thesis is to contribute to understanding the mechanism of generation and stability of such nanoclusters, and propose the basis for this method supported by computer simulations and other theoretical considera-

5 tions. Another actually open system in nanotechnology is the fabrication and stability of metal nanowires and carbon nanotubes. Tao and coworkers [18] have developed an electrochemical method to fabricate metal nanowires by electrochemical deposition. At the same time by electrochemical etching one could fabricate electrodes separated by a nanometer gap. Inspired in these experimental results and observation we have performed simulations in order to gain inside on the foundation of the method. This thesis is divided into four parts: in (Part I) a short introduction to solid state physics and surface structures will be presented, in (Part II) a review of the simulation techniques used in the work are explained together with some technical points, in (Part III) the simulation results concerning to electrochemical nanostructuring are presented in three chapters, in (Part IV) we present a new simulation technique developed as part of this thesis in which a quasi-grand canonical Molecular Dynamic simulation is applied to metal deposition and crystal growth. Finally some general conclusions are given. Ulm (2004) M. Mariscal

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C HAPTER 1. I NTRODUCTION

Chapter 2 Prerequisites from Solid State Physics 2.1 Crystal Structure At the microscopic level, most materials can be considered as a collection of single crystal crystallites. The surface chemistry of the materials as a whole is therefore crucially dependent upon the nature and type of surfaces exposed on these crystallites. Therefore we can understand the surface properties of any material if we know the amount of each type of surface exposed, and have detailed knowledge of the properties of each and every type of the surface plane. It is therefore important that we can independently study different, well-defined surfaces. Most metals only exist in one bulk structural form - the most common metallic crystal structures being: • "bcc" body-center cubic • "fcc" face-center cubic • "hcp" hexagonal close packed

8

C HAPTER 2. P REREQUISITES

FROM

S OLID S TATE P HYSICS

For each of these crystal systems, there are an infinite number of possible surfaces which can be exposed. In practice, however, only a limited number of planes (predominantly the so-called "low-index" surfaces) are found to exist in any significant amount and we will concentrate our attention on these surfaces, in particular for the fcc lattice.

2.1.1 Surface structure of fcc metals Many of the technologically most important metals in surface science and electrochemistry possess the fcc structure: for example the catalytically important metals (Pt, Rh, Pd) all exhibit an fcc structure. The low index faces of this system are the most commonly studied of surfaces, and therefore are the most commonly observed in the present work.

The fcc (100) surface The (100) surface is that obtained by cutting the fcc metal parallel to the front surface of the fcc cubic unit cell. The figure 2.1 a) shows the conventional birdseye view of the (100) surface.

Figure 2.1: fcc surface planes, a) (100), b) (110) and c) (111)

2.1 C RYSTAL S TRUCTURE

9

The fcc (110) surface The (110) surface is obtained by cutting the fcc unit cell in a manner that intersects the x and y axes but not the z-axis. Figure 2.1 b) shows the conventional view of the (110) surface, emphasising the rectangular symmetry of the surface layer atoms. It is clear from this view that the atoms of the topmost layer are much less closely packed than on the (100) surface, in one direction the atoms are in contact, but in the orthogonal direction there is a substantial gap between the rows. This means that the atoms in the underlying second layer are also, to some extent, exposed at the surface, as can be observed in figure Figure 2.1 b). The fcc (111) surface The (111) surface is obtained by cutting the fcc metal in such a way that the surface plane intersects the x−, y− and z− axes at the same value. Figure 2.1 c) shows the conventional birds-eye view of the (111) surface emphasising the hexagonal packing of the surface layer atoms. Since this is the most efficient way of packing atoms within a single layer, they are said to be "close-packed".

2.1.2 Crystal growth With the use of adequate experimental techniques a closer inspection of any surface reveals the presence of irregularities or "defects". A typical type of surface defect is a step between two otherwise flat layers of atoms called terraces (Fig. 2.2). A step defect might itself have defects, for it might have kinks. Different nucleation and growth mechanisms have been found, particularly with modern in-situ probe microscopy techniques [4]. The first step of the deposition of a metal Me start with the formation of metal adatoms. As Lorenz and Staikov present in their review work, the interaction between Me and the substrate

10

C HAPTER 2. P REREQUISITES

FROM

S OLID S TATE P HYSICS

surface S (ΦM e−S ) and Me with Me (ΦM e−M e ), as well as the crystallographic misfit f Me-S are the most important parameters determining the subsequent phase formation and growth mechanism. Three extreme cases can be distinguished at near equilibrium conditions (Fig. 2.3). When: • ΦM e−S  ΦM e−M e 3D phase formation takes place regardless of the M e −

S misfit according to the Volmer-Weber or island growth mechanism in the overpotential deposition (OPD) range

η = E − EM e/M ez+ < 0

(2.1)

where E and EM e/M ez+ denote the actual and the Nernstian equilibrium potential, respectively. • ΦM e−S  ΦM e−M e and dM = dS The formation of well ordered 2D M e adlayers on S accurs in the underpotential deposition (UPD) range.

∆E = E − EM e/M ez+ > 0

(2.2)

Figure 2.2: Different types of defects in real crystal surfaces. 1) terrace, 2) monoatomic step, 3) kink and 4) terrace vacancy

2.1 C RYSTAL S TRUCTURE

11

In systems with strong M e − S interaction, the structure of the well-ordered

2D UPD M e adlayers is determined by the structure matrix and the lateral interaction energy between M e adatoms ΦM e−M e . At relatively low misfit, the Frank-van der Merwe mechanism (layer-by-layer growth) is expected to operate Fig. (2.3).

• ΦM e−S  ΦM e−M e and dM 6= dS At relatively high misfit the StranskiKrastanov mechanism (3D island formation on predeposited 2D M e layers)

takes place. The crystrallographic misfit f is defined by

f=

dM e − d S dS

(2.3)

where dM e and dS denote the atomic-nearest neighbor distances of 3D M e and 3D S bulk phases, respectively.

12

C HAPTER 2. P REREQUISITES

FROM

S OLID S TATE P HYSICS

Figure 2.3: Schematic representation of different crystal growth modes. a) Volmer-Weber, b) Frank-van der Merwe, c) Stranski-Krastanov.

Part II

Chapter 3 Simulation Techniques 3.1 Molecular Dynamics Molecular Dynamics (MD) is a computer simulation technique that allows one to predict the time evolution of a system of interacting particles. MD follows the dynamics of a single system and produces averages of the form : M

1X Ai (r N ) < A >= M i=1

(3.1)

where A is any equilibrium average quantity, r N represents the coordinates of all N particles and M is the number of measurements made as the system evolves.

The basic idea is simple and we will enter directly to the scheme to understand the technique. First for a system of interest, one has to specify: • a set of initial conditions: (initial positions and initial velocities of all particles in the system).

• Interaction potentials for deriving the forces among all the particles. Second, the evolution of the system in time can be followed by solving a set of classical equations of motion for all particles in the system. Whitin the framework

16

C HAPTER 3. S IMULATION T ECHNIQUES

of classical mechanics, the equations that govern the motion of classical single particles are the ones that correspond to the second law of classical mechanics formulated by I. Newton over 300 years ago: ~i mi ~ai = F mi

(3.2)

d2~r ~i =F dt2

~ i , and ~ri are the acceleration, Where mi is the mass of the i − th particle, ~ai , F force and position respectively. If the particles of interest are atoms, and if there

are a total of N of them in the system, the force acting on the ith atom at a given time can be obtained from the interatomic potential V(r1 , r2 , r3 , . . . ., rN ) that, in general, is a function of the positions ri of all the atoms: ~ i = −∇V(r ~ F i)

(3.3)

Once the initial conditions and the interaction potential are defined, the equations of motions can be solved numerically. The solutions are the positions and velocities of all the atoms as a function of time, ~ri (t), ~vi (t). To integrate Newton’s equations of motion there are many procedures, one of the most efficient is the so-called predictor-corrector algorithm. This method consist of three stages: 1) Predictor, 2) Force calculation and 3) Corrector. But, let us define first the following variables: q1 (t) = hdr(t)/dt, q2 (t) = h2 /2d2 r(t)/dt2 , q3 (t) = h3 /6d3 r(t)/dt3 , q4 (t) = h4 /24d4 r(t)/dt4 , q5 (t) = h5 /120d5 r(t)/dt5 to be used in the Taylor expansion, where h is the time step. • Predictor: from the positions and their time derivatives up to a certain order

(all at time t) one "predicts" the same quantities at time t+h using the Taylor expansion:

3.1 M OLECULAR DYNAMICS r(t + h) = r(t) + q1 (t) + q2 (t) + q3 (t) + q4 (t) + q5 (t)

17 (3.4)

q1 (h + t) = q1 (t) + 2q2 (t) + 3q3 (t) + 4q4 (t) + 5q5 (t) q2 (h + t) = q2 (t) + 3q3 (t) + 6q4 (t) + 15q5 (t) q3 (h + t) = q3 (t) + 4q4 (t) + 10q5 (t) q4 (h + t) = q4 (t) + 5q5 (t) q5 (h + t) = q5 (t) • Force Calculation: the force acting on a given particle is computed for the

predicted positions. The acceleration ~a = F~ /m will be in general different from the "predicted acceleration", therefore the difference between the two

constitutes an error signal. F (t + h) 2 h − q2 (t + h) (3.5) 2m • Corrector: the error signal is used to correct the positions and their derivates. δq2 (t + h) =

All the corrections are proportional to the error signal. The coefficients of proportionality are "magic numbers" chosen to maximize the stability of the algorithm. r c (t + h) = r(t + h) + c0 δq2 (t + h)

(3.6)

q1c (t + h) = q1 (t + h) + c1 δq2 (t + h) q2c (t + h) = q2 (t + h) + c2 δq2 (t + h) q3c (t + h) = q3 (t + h) + c3 δq2 (t + h) q4c (t + h) = q4 (t + h) + c4 δq2 (t + h) q5c (t + h) = q5 (t + h) + c5 δq2 (t + h) where c0 , c1 , c2 , c3 , c4 , c5 are 3/20, 251/360, 1, 11/18, 1/6 and 1/60 respectively At this time it is important to mention the limitations of the classical MD technique, one of the most important is that the electrons are not presented explicitly, they are introduced through the potential energy surface, that is a function

18

C HAPTER 3. S IMULATION T ECHNIQUES

of atomic positions only. Also the potential energy is approximated by an analytic function that gives the potential energy as a function of coordinates. The availability of a good potential function is the one of the main conditions for MD simulations and we will discuss in following sections.

3.2 Monte Carlo Monte Carlo methods is a common name for a wide variety of stochastic techniques. These techniques are based on the use of random numbers and probability statistics to investigate problems in areas as diverse as physics, biology, chemistry and economics. Monte Carlo methods are distinct from molecular dynamics method but can also sample the phase space of the system and, hence, an appropiate for calculating thermodynamic quantities or for performing simulated annealing calculations. Unlike the molecular dynamics method it cannot be used to study time-dependent properties (except Kinetic Monte Carlo) but it does have other feactures that are advantageous in some circumstances. Starting from the classical expression for the partition function Z with volume (V ), temperature (T ) and number of particles (N ) constant (N V T or canonical ensemble): ZN V T = c

Z

dpN dr N exp[−H(r N pN )/kB T ]

(3.7)

where r N represent the coordinates of all N particles, and pN the momenta. The function H(r N pN ) is the Hamiltonian of the system. From which we can obtain

the total energy of an isolated sytem as H(r, p) = K(p) + V(r), where K is the

kinetic energy and V is the potential energy. Finally c is a constant of proportion-

ality. Now, if we consider the average value of some observable A [1]: R N N dp dr A(pN , r N ) exp[−βH(pN , r N )] R < A >N V T = dpN dr N exp[−βH(pN , r N )]

(3.8)

3.2 M ONTE C ARLO

19

where β = 1/kB T . In this equation the observable A has been expressed as a

function of the coordinates and momenta. The difficult problem in eq. (3.8) is the

computation of averages of functions A(r N ), the computation of A(pN ) is trivial

because of the quadratic dependence of the kinetic energy on the momenta. To

solve this problem we will use the Monte Carlo method, or more precisely, the Monte Carlo importance sampling algorithm.

3.2.1 Importance Sampling Before discussing importance sampling, let us first look at the simplest Monte Carlo technique, that is, random sampling. Suppose we wish to evaluate numerically a 1D integral I: I=

Z

b

(3.9)

dxf (x) a

equation (3.9) can be rewritten as I = (b − a) < f (x) > where < f (x) > denotes the unweighted average of f (x) over the interval [a, b]. In brute force

Monte Carlo, this average is determined by evaluating f (x) at a large number L

of x values randomly distributed over the interval [a, b]. It is clear that as L → ∞,

this procedure should yield the correct value for I. However this method is of little

use to evaluate averages such as in equation (3.8) because most of the computing is spent on points where the Boltzmann factor is negligible. It would be much preferable to sample many points in the region where the Boltzmann factor is large and few elsewhere. This is the basic idea behind importance sampling. Now, suppose we wish to compute the definite integral in equation (3.9) by Monte Carlo sampling, but with the sampling points distributed nonuniformly over the interval [a = 0, b = 1], according to some nonnegative probability density w(x). Clearly, we can rewrite equation (3.9) as : Z 1 f (x) I= dxw(x) w(x) 0

(3.10)

20

C HAPTER 3. S IMULATION T ECHNIQUES

Let us assume that we know that w(x) is the derivative of another (nonnegative, nondecreasing) function u(x), with u(0) = 0 and u(1) = 1. Then I can be written as: I=

Z

1

du 0

f [x(u)] w[x(u)]

(3.11)

The next step is to generate L random values of u uniformly distributed in the interval [0, 1]. We then obtain the following estimate for I: I≈

L

1 X f [x(ui )] L i=1 w[x(ui )]

The variance (σI2 ) can be estimate as [1]: "*  +   # 2 2 f f 1 2 σI = − L w w

(3.12)

(3.13)

Equation (3.13) shows that the variance in I still goes as 1/L, but the magnitude of this variance can be reduced greatly by choosing w(x) such that f (x)/w(x) is a smooth function of x. As the integrand in equation (3.8) is nonzero only for those configurations where the Boltzmann factor is nonzero, it would clearly be advisable to carry out a nonuniform Monte Carlo sampling of configuration space, such that the weight function w is approximately proportional to the Boltzmann factor. Unfortunately, this scheme cannot be used to sample multidimensional integrals such as equation (3.8). The reason is simply that we do not know w.

3.2.2 The Metropolis algorithm The previous section suggest that it is not possible to evaluate integrals such as equation (3.8) by direct Monte Carlo sampling. However, in many cases, we are not interested in the configurational part of the partition function itself but in averages of the type < A >N V T =

R

dr N exp[−βV(r N )]A(r N ) R dr N exp[−βV(r N )]

(3.14)

3.2 M ONTE C ARLO

21

Hence, we wish to know the ratio of two integrals. What Metropolis points out is that it is possible to devise an efficient Monte Carlo scheme to sample such a ratio. We denote the configurational part of the partition function by Z ZN V T = dr N exp[−V(r N )/kB T ]

(3.15)

The ratio of the exponent is the probability density to find the system in a configuration around r N . Let us denote this probability density by N (r N ) =

exp(−V(r N )/kB T ) ZN V T

(3.16)

Suppose now that we are somehow able to randomly generate points in configurational space according to this probability distribution. This means that, on average, the number of points ni generated per unit volume around a point r N is equal to LN (r N ), where L is the total number of points that we have generated.

Let us next consider how to generate points in configurational space with a

relative probability proportional to the Boltzmann factor. The general approach is first to generate a configuration r N , which has a Boltzmann factor exp[−βV(rold )]. 0

Next, we generate a new trial configuration r N , by adding a small random displacement ∆ to the old configuration. The Boltzmann factor of this trial configuration is exp[−βV(rnew )]. We must now decide whether we will accept or reject the trial configuration. Many rules for making this decision satisfy the constraint that on average the probability of finding the system in a configuration r new is proportional to N (new). Let us now derive the Metropolis scheme to determine the

transition probability π(o → n) to go from configuration old(o) to new(n). The

matrix elements π(o → n) must satisfy one obvious condition: in equilibrium, the average number of accepted trial moves leaving state o must be exactly equal

to the number of accepted trial moves from all others states n to state o. A much stronger condition is the detailed balance condition that implies the following: N (o)π(o → n) = N (n)π(n → o)

(3.17)

22

C HAPTER 3. S IMULATION T ECHNIQUES

We denote the transition matrix that determines the probability to perform a trial move from old → new by α(o → n), where α is usually referred to as the underlying matrix of the Markov chain. The next stage is the decision to either accept

or reject this trial move. Let us denote the propability of accepting a trial move by acc(o → n). Clearly, (3.18)

π(o → n) = α(o → n) × acc(o → n)

In the original Metropolis method, α is chosen to be a symmetric matrix [acc(o →

n) = acc(n → o)], however there are examples where is not symmetric. If α is symmetric, we can rewrite equation (17) in terms of acc(o → n): N (o) × acc(o → n) = N (n) × acc(n → o)

(3.19)

acc(o → n) N (n) = = exp{−β[V(n) − V(o)]} acc(n → o) N (o)

(3.20)

and therefore,

The choice of Metropolis for acc(o → n) is:

acc(o → n) = N (n)/N (o) if N (n) < N (o) acc(o → n) = 1 if N (n) > N (o)

In summary, the Metropolis Monte Carlo algorithm works as follow:

1. Choose the initial configuration, calculate the energy 2. Make a "move" (for example, pick a random displacement), calculate the energy for new "trial" configuration. 3. Decide whether to accept the move: if V(n) − V(o) < 0, then accept the new configurarion, if V(n) − V(o) > 0, then calculate acc(o → n) = exp{−β[V(n) − V(o)]} draw a random number [0,1] then if acc(o → n) >

configuration, otherwise, stay at the same place.

then accept the new

3.2 M ONTE C ARLO

23

4. Repeat from step 2, accumulating sums for averages.

3.2.3 Grand-Canonical ensemble In the ensemble that we have discussed above the number of particles N , the volume V , and the temperature T are kept constant. However we are interested in many cases in electrochemical problems, especially in metal depostion and dissolution. For this kind of study one of the best simulation techniques is the Monte Carlo method in the grand canonical (GC) ensemble, in which the chemical potential µ, the volume V , and the temperature T are fixed. In the GC ensemble the temperature and chemical potential are imposed and the number of particles is allowed to fluctuate during the simulation. This makes these simulations very different from the conventional ensembles, where the number of particles is fixed during the simulation. The distribution function in this case is given by: exp(µβN ) exp[−Hβ(pN , r N )] R N N N N N =0 exp(µβN ) dp dr exp[−Hβ(p , r )]

ρµV T = P∞

(3.21)

and the average value of any observable A will be:

< A >µV T =

R N N exp(µβN ) p dr A(pN , r N ) exp[−Hβ(pN , r N )] N =0 R N N (3.22) P∞ dp dr exp[−Hβ(pN , r N )] N =0 exp(µβN )

P∞

Let us now consider the trial moves in order to execute the GC ensemble within the Metropolis formalism: • Displacement of particles: A particle is selected at random and a new conformation is given by a small random displacement. This move is accepted with a probability:

Wj→i = min(1, exp −(Vij )/kB T )

24

C HAPTER 3. S IMULATION T ECHNIQUES where Vij is the potential energy change associated with the motion of the particle. • Insertion of particles: An attempt is made to insert a particle at a random position in the simulation box. The new configuration is accepted with the probability:

WN →N +1 = min(1,

V δ 3 (N

+ 1)

exp((µ − ∆VN +1,N )/kB T ))

where V is the volume where the particles are created. δ =

p (h2 /2πmkT )

is the thermal De Broglie wavelength and VN +1,N = VN +1 − VN is the

potential energy change associated with the creation of a particle.

• Removal of particles: A particle is chosen at random and a removal attempt is accepted with the probability:

WN →N −1 = min(1,

δ3N exp((−µ − ∆VN −1,N )/kB T )) V

where VN −1,N = VN −1 − VN is the potential energy change associated with

the removal of a particle.

In order to get a good efficiency for the grand canonical Monte Carlo algorithm shown above, the trial moves were modified. The adatoms can not be added in any point of the simulation box, they can be added or deleted from the system only in a subspace defined by a collection of spheres disposed in a mesh as shown figure 3.1. This simulation trick is know as "configurational-bias", therefore the factor of acceptance must be modified, in this case the volume V is now the volume occupied by the total number of spheres.

3.3 I NTERATOMIC P OTENTIALS

25

Figure 3.1: Schematic representation of the Bias-GC Monte Carlo used in the simulations, a) used for growth the nanowire between a STM and a surface. b) to study the stabiliy of metal clusters.

3.3 Interatomic Potentials In order to use Molecular Dynamics or Monte Carlo methods we have to define the rules that are governing interactions of atoms in the system. In classical simulations these rules are often expressed in terms of potential functions. The potential function V(r1 , r2 , . . . ., r N ) describes how the potential energy of a system of N atoms depends on the coordinates of the atoms1 .

How to obtain the potential function for a particular system ? One can assume a functional form for the potential function and then choose 1

It is assumed that the electrons adjust to new atomic positions much faster than the motion of

the atomic nuclei (Born-Oppenheimer approximation)

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the parameters to reproduce a set of experimental data, this gives so-called empirical potential functions (e.g. Lennard-Jones, Morse, etc). On other hand one can calculate the electronic wavefunction for fixed atomic positions. This is difficult for a system of many atoms. Different approximations are used and analytic semi-empirical potentials are derived from quantummechanical arguments (e.g. Embedded atom method, Glue model, Bond-order, etc). One can perform direct electronic structure calculations of forces during socalled ab-initio MD simulations (e.g. Carr-Parrinello method). When choosing potentials we have to consider that the potential reproduce properties of interest as closely as possible, that they can be used to study a variety of properties to which it was not fitted, and also a important point is the computational time consumed by the calculation of the force in the case of MD simulations.

3.3.1 Pair potentials Pair potentials as Lennard Jones, and Morse, are used for inert gases, intermolecular van der Waals interaction in organic materials, or for investigation of general classes of effects (without importance of the material). The total potential energy of the system of N atoms interacting via pair potentials is: V(r1 , r2 , . . . , r N ) =

XX i

j>i

V2 (rij )

(3.23)

where rij = |~rj −~ri |. As can be seen in equation (21) the interaction of any pair of

atoms depends only on their spacing and is not affected by the presence of other atoms.

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27

Limitations of pair potentials: Many body effects are important in real materials. Pair potentials do not depend on the enviroment, and therefore cannot take into account surfaces defects, and the most important surfaces problems. There are many parameters of metals that are overestimated by pair potentials (e.g. vacancy formation energy, melting temperature, elastic constants of solids, etc.) and therefore pair potentials are not adequate for describing metallic systems.

3.3.2 Many body potentials An alternative simple but rather realistic approach to describe the bonding on metallic systems is based on the concept of local density. This allows one to account for the dependence of the strength of individual bonds on the local environment which is especially important for simulations of surfaces and defects, as will be shown in the following chapters. There are many methods proposed to solve this problem, but mainly they are all the same in spirit; the most popular is the embedded atom method (EAM) developed by Daw and Baskes [5] but the effective medium theory [7], FinnisSinclair [8] and the Glue model [9] are also often used. As was mentioned above, all of these methods are based on different physical arguments, but result in a similar expressions for the total energy of the system of N atoms: Etot =

P i

Vi = Fi (ρi ) + where ρi =

P

Vi

(3.24)

1X φij (rij ) 2 j6=i

fj (rij ). Interpretation and functional form of F , f , and φ depend

j6=i

on the particular method. From the point of view of effective medium theory or

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the embedded atom method the energy of the atom i is determined by the local electron density at the position of the atom and the function f describes the contribution to the electronic density at the site of atom i from all atoms j. The sum over the function f is therefore a measure of the local electron density ρ i . The embedding energy F is the energy associated with placing an atom in the electron environment described by ρ. The pair potential term φ describes electrostatic contributions. The general form of the potential can be considered as a generalization of the basic idea of the density functional theory (the local electron density can be used to calculate the energy). The main advantage of these methods over pair potentials is the ability to describe the variation of the bond strength with coordination. Increase of coordination decrease the strength of each of the individual bonds and increase the bond length. In order to use this potential in Molecular Dynamic simulations we need to find the forces: ~ ~r Etot = − F~i = −∇ i

X  ∂Fi (ρi ) ∂fj (rij ) j6=i

∂ρi

∂rij

 ∂Fj (ρj ) ∂fi (rij ) ∂φij (rij ) (~ri − ~rj ) + + ∂ρj ∂rij ∂rij rij (3.25)

Only inter-particles distances rij are needed to calculate energy and forces. To define an EAM potential we have to define three functions, the embedding function F (ρ), the pair potential φ(rij ), and the electron density function f (rij ). Foiles, Daw, and Baskes only give the data as a set of points and then use spline interpolation to get each term efficiently. R. Johnson [54] has developed an analytic EAM introducing a smooth cutoff function for the electron density function and the two-body potential, and an analytic form of the embedding energy function. We have used both EAM, in early simulation on the generation of metal nanoclusters with the aim of a STM tip, we have used EAM potentials provided by Foiles, Daw and Baskes, however, in the develepment of the new model to

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29

study metal deposition we have used the analytic EAM developed by Johnson. Alloy EAM potentials can be constructed from elemental EAM potentials if the potentials are normalized and unified cutoff functions are used. To fit such an EAM potential set, the generalized pair potentials were chosen to have the form r

r

Ae−α( re −1) Be−β( re −1) φ(rij ) = − 1 + ( rre − κ ) 1 + ( rre − λ )

(3.26)

where re is the equilibrium spacing between nearest neighborns, A, B, α, β are four adjustable parameters, and κ, λ,  are three additional parameters for the cutoff. 0,5 φ(r) dφ/dr

0,4

φ [eV] and dφ /dr [eV/Å]

0,3 0,2 0,1 0 -0,1 -0,2

2

4

3

5

6

r [Å]

Figure 3.2: Pair potential functions and their derivatives The electron density function is taken with the same form as the attractive term in the pair potential with the same values of β, and λ, r

fe e−β( re −1) f (r) = 1 + ( rre − λ )

(3.27)

  1 f b (r) aa f a (r) bb φ (r) = φ (r) + b φ (r) 2 f a (r) f (r)

(3.28)

Using the alloy model, the pair potential between different species a and b is constructed as

ab

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C HAPTER 3. S IMULATION T ECHNIQUES To have embedding energy functions that can work well over a wide range of

electron density, Johnson uses three equations to separately fit to different electron density ranges, ρ < ρn , ρn 6 ρ < ρo and ρo 6 ρ. By using ρn = 0.85ρe and ρo = 1.15ρe where ρe is the equilibrium electron density, we can ensure that all equilibrium properties can be fitted in the electron density range ρ n 6 ρ < ρo . For a smooth variation of the embedding energy, these equations are required to match values and slopes at their junctions. 1 f(r) df/dr

f [eV/Å] and df /dr [eV/Å^2]

0,75 0,5 0,25 0 -0,25 -0,5 -0,75 -1

1

2

4

3

5

6

r [Å]

Figure 3.3: Electron density functions and their derivatives These equations are listed as follows:  i 3 X ρ F (ρ) = Fni − 1 , ρ < ρn , ρn = 0.85ρe ρn i=0

i ρ − 1 , ρn 6 ρ < ρo , ρo = 1.15ρe F (ρ) = Fi ρ e i=0   η    η ρ ρ . F (ρ) = Fe 1 − ln , ρo 6 ρ ρe ρe 3 X



(3.29)

(3.30) (3.31)

With this model, the parameters needed to define all the metals are listed in Appendix A. The functions and their derivatives for Johnson’s EAM with param-

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31

eters for Au are displayed in Figures: 3.2, 3.3 and 3.4

1 F(ρ) dF/dρ F(ρ) [eV] and dF/dρ [eV/Å]

0

-1

-2

-3

-4

0

10

20

30

40

50 ρ [eV/Å]

60

70

80

90

100

Figure 3.4: Embedded functions and their derivatives A three-dimensional plot of the potential energy surface when a single atom is moved on the surface is shown in figure 3.5. The energy was calculated for all configurations of the system with the adatom placed on each point along the fine 10 Å x 10 Å grid consisting of 100 x 100 points. In figure 3.5-a) the generated 3-D potential energy surface is shown, and in frame b) a 2-D contour plot is illustred, we can perfectly note hollow, bridge and top sites for adsorption.

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Figure 3.5: Potential energy surface for an Au ad-atom placed on Au(111) generated with the embedded atom method a) 3D view and b) 2D contour plot

Part III

Chapter 4 Nanodecoration of surfaces with a STM 4.1 Introduction The invention of the Scanning Tunnelling Microspopy (STM) by Binning and Rohrer in 1981 was a revolution in surface science, and after five years they were awarded with the Nobel Prize. In a STM a very sharp metallic tip mounted upon a set of piezoelectric transducers is supported very close to the surface of the sample to be studied. If a potential difference is applied between the tip and the surface and the tip-surface separation ∆D, is small enough then electrons can pass between them by quantum mechanical tunnelling. The tunnelling current is a very rapid (exponential) function of ∆D, therefore a very sensitive probe of this separation. Imaging of the surface topology may then be carried out in one of two ways: 1) in constant height mode, in which the tunnelling current is monitored as the tip is scanned parallel to the surface. 2) In constant current mode, in which the tunnelling current is maintained constant as the tip is scanning across the surface. Therefore, images with atomic

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resolution can be obtained with a STM. However in the past few years scanning tunnelling microscopy was not used only as a technique to image surfaces in real space, it was also used as a tool for positioning single atoms and molecules on surfaces with a high precision. In other words, we can imagine a STM tip as an atomic pencil with which we can write in atomic dimensions, manipulating atom by atom if we would like, contemplating the dream of the physicist Richard Feynman in 1959. Kolb and co-workers [25] developed an accurate technique to generate small clusters on an electrode surface by means a STM tip. The technique is known as "Tip Induced Metal Deposition (TIMD)", and a schematic representation about how it works is shown in figure 4.1. The STM tip and the working electrode are immersed in a solution containing M +z ions. At first (I), the electrode potential is held at a value a little above the deposition potential φ0 for bulk M . Under these conditions the electrode surface is already covered by a monolayer of M , a phenomenon known as underpotential deposition (UPD), and the metal M +z is deposited onto the tip, as shown II).

Figure 4.1: Schematic representation of "Tip Induced Metal Deposition (TIMD)" The STM tip (III) is then brought into contact with the electrode surface for a short period of time, during which a so-called jump-to-contact accurs. Finally

4.1 I NTRODUCTION

37

by retraction of the STM tip (IV) the connective neck breaks and a small metal cluster is left behind on the surface. This method can be automatizated to produce complex structures that can be used in future nanotechnology. This method has been found by Kolb and co-workers to produce clusters for a number of systems, but it fails in others.

Figure 4.2: STM image of an array of Pd clusters on Au(111) taken from [27] As an example Fig. 4.2 shows a matrix of Pd clusters supported on a substrate surface of Au(111). It can be appreciated that the technique has a great reproduction and permits to control the size of the clusters formed with great accuracy. In this chapter we will see by means molecular dynamics the generation of small metal clusters by TIMD. The process to generate these clusters can be divided in two stages, determined by two scales of time (in the electrolyte concentration used in the experiments). The first stage we call "mechanical stage", and involves the mechanical formation of the connecting neck between the STM tip and the electrode surface, identation or penetration of the tip into the surface, retraction

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of the tip, elongation and finally the mechanical rupture of the connecting neck. The second stage we call "chemical stage" and involves the exchange of material of the cluster with the electrolyte, and we will see in the next section. In this section we will see the mechanical stage and we will try to answer the following open questions: • Which atomistic processes are present in the cluster formation ? • How is the dependence of cluster size with the approach of the STM tip ? • Which systems produce metal clusters and which ones fail ? • What is the composition of the cluster ?

4.2 The Model and calculation details All the results presented in this section were obtained by means of molecular dynamics (MD), employing in all cases a very similar simulation scheme. The MD program employed was the XMD1 , as developed by John Rifkin [39]. The code use the corrector-predictor algorithm to integrate the equation of motion, and we have chosen an integration time step of 2.0 fs in order to ensure energy conservation during adiabatic simulations. A typical initial configuration is shown in figure 4.3, in which a cut in the (Y − Z) direction of the three-dimensional system is shown.

The STM tip consists of a fixed number of Pt atoms (light circles on rigid core), whose positions were fixed rigidly with respect to each other, yielding a hard core on which the atoms of the metal M (dark circles) to be deposited on the surface were attached. The substrate (bottom light circles) was emulated by 1

XMD is a free software designed for performing molecular dynamics simulations of metals

and ceramics)

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Figure 4.3: Typical initial configuration employed in TIMD. The section in the (Z − Y ) direction is shown. Light gray circles represent Pt atoms belonging to the

STM tip. Darl gray circles denote the atoms of the material M being deposited and the light gray circles (down) indicate the substrate atoms. Note also the presence of the UPD monolayer in this case.

means of 8 fcc(111) lattice planes made of 400 atoms each, with cyclic boundary conditions operating in the X − Y plane. The two lower planes were held fixed at

their equilibrium positions; they provided the force field in wich the remaining 6 lattice planes moved. In some systems, such as for Pb deposition on Au(111), the substrate is known to be covered by a monolayer of the deposit M prior to cluster formation, a phenomenon know as underpotential deposition (UPD); for such

cases, the lattice planes representing the substrate were covered by an additional

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monolayer (UPD ML) of M in our simulation as shown the dark circles on the substrate in Fig. 4.3. Figure 4.3 also displays a few important simulation parameters: • zt = Average position of the top atoms(s) of the rigid tip • zs = Average position of the atoms belonging to the first surface lattice plane of the substrate (or the monolayer, if this is present) located at the

corners of the simulation cell • dts = zt − zs • d0 = Distance between the lowest atom in the tip and the first surface lattice plane at the beginning of the simulation.

The geometrical shape of the STM tip (labeled as "Pt rigid core" in Fig. 4.3 was taken as semiellipsoidal, and obtained by sectioning a piece of the Pt fcc bulk structure. Two types of structures were employed for this tip in the present work. These were: 111 - fcc crystalline, with the (111) face parallel to the surface of the substrate 110 - fcc crystalline, with the (110) face parallel to the surface of the substrate, and (111) facets on the sides. In the case of [111]-type tips, an ellipsoidal portion of M was deposited in the rigid Pt core, melted by heating the system and then gently quenchend to reach the simulation temperature. The atoms spontaneously arranged into a fcc lattice with the (111) face parallel to the substrate surface. In the case of [110]-type tips, a suitable portion of the Pt atoms were replaced by M atoms, in order to represent the deposit M on the tip which is partially transferred during cluster formation.

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The interaction between the particles were calculated from the embedded atom method (EAM) which was discussed in section 3.3.2, with a spherical cutoff radius rc . For the adsorbate-substrate systems Pd-Au, Cu-Pt and Ag-Pt, the potentials employed were those devised by Foiles et al. [5] with rc = 6.0 Å . In the case of the system Pb-Au the potential employed was with the parameters reported by Imeson and De’ Bell [6] with rc = 5.6 Å . At the beginning of the simulation the STM tip covered with M was positioned above the substrate at a distance d0 > rc . The system was equilibrated during 40 ps, occasionally scaling the velocities to reach a final average temperature close to 300 K. The approach and subsequent retraction motion of the tip was simulated by stepping the Z position of the Pt atoms, and consequently the tip-surface distance, in steps of 0.006 Å every 2 ps. The motion was reversed at a distance, d ca , of closest approach. This resulted in an average velocity of 30 cm/s. The d ca values reported here are referred to jump-to-contact distance, wich means d ca = 0 for the configuration where the tip has come right in contact with the surface, forming a connecting neck. Depending on dca , the whole approach-retraction cycle took between 5 and 10 ns, values considerably shorter than the corrensponding experimental times, which are of the order of ms.

4.2.1 The system Pd on Au(111) The system Pd on Au(111) was studied experimentally by Kolb’s group [27] and was used as a prototype because of the great interest in future applications on catalysis. Recently, our group has studied this system in connection with experiments [32, 33], in collaboration with two research groups from Technischen Universität München - TUM and Universidad de Cordoba. In the following we will see some experimetal observations in comparison with our simulation results.

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Experiments

The experiments presented in this section were performed by Stimming’s group (TUM). In their experiments the Kolb’s method [27] was a little modified, and deposition of palladium onto the tip was made in a separate electrochemical cell containig a 0.1 mM palladium sulphate solution in 0.1 M sulphuric acid. The generation of the clusters and stability measurements were then performed in pure 0.1 M sulphuric acid. This procedure permits to work at sample potentials at which electrodeposition of palladium onto the substrate would otherwise occur. They wanted to investigate which effect a variation of the parameters for the tip approach has on the generated clusters. Technically, this approach was realized by adding a short voltage pulse to the z-piezo voltage of the STM. The clusters generated varied in height between 0.3 to 1.8 nm . Their diameter was found to be between 5 and 20 nm; however, this values are influenced by the folding of the shape of the cluster with the shape of the tip and are therefore somewhat uncertain. But there is definitely a tendency that particles smaller in diameter are less high. The dependence between tip approach and cluster height is shown in Fig. 4.4; negative values of z denote an approach towards the surface. Each point in the plot results from averaging over four to nine clusters, the vertical error bars are standard deviation from this average.

From a certain threshold on, the height of the clusters increase monotonously with decreasing distance of closest approach. The threshold lies approximately between -0.5 and -0.6 nm, and should to be close to the point where the jump to contact occurs.

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Figure 4.4: Height of Pd clusters on Au(111) as a function of distance of closest approach during generation. (From experiments of Stimming’s group [32, 33]) Simulations - Cluster generation We have performed a serie of 12 simulations2 at different closest approach distance dca , in which we have used a Au(111) surface composed by eigth layers. A typical starting configuration is shown in figure 4.5-a). A series of snapshots taken during the generation of a typical cluster is shown in figure 4.5, in which a closest approach distance between the tip and the surface was dca = −6.37 . In frame b) the jump to contact shows that for this system the material displacement is from the tip to the surface, we also can note that the

contact point is rather symmetrical with respect to the z-axis along the STM tip. In fact, the direction in which the material displaced can be correlated with the 2

In collaboration with Mario G. Del Popolo, Facultad de Cs. Quimicas, Universidad Nacional

de Cordoba. (Some pictures presented for this system were edited by Mario Del Popolo)

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Figure 4.5: Snapshots taken during the generation of a Pd cluster on Au(111). a)Initial state of the system, dca = +2.24 ; b) jump-to-contact, dca = 0.0 ; c) closest approach distance, dca = −6.37 ; d) final stage after retraction of the tip, dca = −11.04 cohesive energy of both materials, for example in this case the cohesive energy of Pd (3.91 eV) is a little smaller than Au (3.93 eV). In frame c) the closest approach distance is shown, note how some Au atoms from the surface appear on the connecting neck. The resulting cluster after withdrawn the tip is shown in frame d), a 180 atoms cluster was obtained, constituted of 158 Pd and 22 Au atoms. Figure 4.6 shows the size and the composition of the clusters as a function of the distance dca that the tip was moved towards the surface after jump to contact. A small tip movement resulted in small clusters that were composed of palladium only. With increasing distance dca that the tip is moved toward the surface, the clusters growth larger and contain an increasing amount of gold atoms that

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Figure 4.6: Number of atoms in the clusters as a function of closest approach distance dca , the numbers close to the points give the percentage of Au atoms in the cluster.

have been detached from the electrode surface. We also note that when the identation of the tip was stronger some Pd atoms were found in the first layers of the substrate surface, and some Au atoms from the first layers were attached to the end of the STM tip. A set of clusters taken immediately after breaking of the connecting neck are illustrated in figure 4.7, as can be observed, the clusters are the larger the further the tip has been moved towards the substrate. At the same time their gold content becomes higher. The newly formed clusters have the shape of a truncated pyramid. However, it is to be borne in mind that in the experiments a comparatively long time, of the order of seconds, passes between the generation and the observation of the clusters, and that in the meantime their shape may change either through re-

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arrangement or through ion exchange with the solution. This point will be further analyzed below.

Figure 4.7: Pd clusters generated on a Au(111) surface. Dark and light spheres denote Pd and Au atoms respectively. The closest approach distance d ca were: a) -0.36 Å, b) -3.96 Å, c) -5.16 Å, d) -6.37 Å,e) -7.58 Å, f) -8.78 Å. Finally a remarkable characteristic of the nanostructuring with Pd clusters, is that the electrode surface presents practically no geometrical damage. Simulations - Cluster stability Experiments of clusters generated on a bare Au(111) surface showed that when the potential was stepped to a value where the clusters dissolved, the height of the clusters did not change continuosly, rather a stepwise fashion was observed. This leads to the conclusion that the cluster dissolution takes place simultaneously at the borders of all the layers. Since Pd and Au form stable alloys, alloyed clusters should be more stable than

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pure Pd ones. This idea was analyzed by means Monte Carlo simulations, choosing the grand canonical ensemble because we can control the chemical potential of the metal to be deposited as we have seen in chapter 3.

Figure 4.8: Evolution of the number of atoms in a cluster as a function of the number of MCS at different chemical potentials µ. a) Pure Pd cluster b) Alloyed Au-Pd cluster with 253 atoms (15.3 % Au).

We have taken the cluster shown in figure 4.7-f) with 253 atoms, and performed MC simulations at different chemical potentials in order to investigate the stability of such clusters. In order to compare the stabiliy of the alloyed cluster with that of the pure Pd cluster, we have replaced all Au atoms by Pd and a pure

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Pd cluster was obtained. Figure 4.8 shows the evolution of the number of atoms (n) on the cluster as a function of the number of Monte Carlo steps (MCS) for both pure and alloyed Au-Pd cluster at different chemical potentials. Both types of clusters start to grow when µ is larger than -3.82 eV, and start to dissolve below this value. However, the alloyed cluster dissolve much more slowly, and reach a constant size after some time. For example, at µ=-4.3 eV a 103 atoms cluster (40 % content of Au) remained stable during all the course of the simulation. From the analysis of the atomic coordinates we have found that the cluster at µ=-3.70 eV grew taking the form of a truncated hexagonal pyramid, whose faces presented (111) and (100) packing. When the chemical potential is only a little below the equilibrium value for Pd, the dissolution proceeds at kink sites from all levels simultaneously. Since stable clusters have a shape of a truncated pyramid, this entails that the height of smaller clusters decrease more rapidly, as was observed in experiments [32, 33].

4.2.2 The system Pb on Au(111) The deposition of three-dimensional aggregates of Pb on Au(111) has been recently undertaken by R. Ullmann [29]; this system presents great discrepancy in the experimental results. The fabrication of Pb clusters was possible on Au(111), though only at potentials very close to the bulk Pb deposition potential, and even under these conditions the clusters could only be briefly observed before their dissolution. Figure 4.9 shows 11 clusters generated with the TIMD method by Ullmann [29], as we can see the clusters are rather irregular in height and size.

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Figure 4.9: STM image of Pb clusters generated on a Au(111) surface with a UPD monolayer, taken from [29] Cluster generation To gain further insight into the fabrication and the stability of these metal clusters we have performed extensive Molecular Dynamics simulations. Our model system is the same to that presented in previous section, but with different metals and geometry. The system consists of a semi-ellipsoidal core tip of fixed Au atoms, from which several layers of mobile lead atoms were suspended. The electrode surface was modeled by eight layers of gold atoms arranged in a fcc(111) geometry; the top six gold layers are mobile, and the last two were fixed in order to simulate the bulk. The initial composition for all cases was: 2247 Au atoms (belonging to the rigid core of the tip), 2137 Pb atoms (all movable, including the UPD monolayer) and 3200 Au atoms (belonging to the substrate surface sample). The interactions between the various atoms were calculated using the embedded atom method, fol-

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lowing the instruction and parameters for lead and gold reported by K’De Bell and Imeson [6]. As mentioned in previous sections, the four important stages in cluster formation are: • Initial configuration with the STM tip at a distance of 6.35 Å from the surface (bigger than the cutoff distance rc = 5.6 Å used for the EAM potential)

• The so-called jump-to-contact, in which the atoms from the tip and the sur-

face jump to form a bond. In the case of Pb/Au(111) the displacement of material is from the tip to the surface.

• Indentation, from which the closest approach distance between the tip and the surface is reached after jump-to-contact.

• Finally, retraction of the tip and rupture of the connecting neck, and a cluster is left behind on the surface.

As mentioned above, one of the most important parameters to control the size of these clusters is the distance to which the tip approaches to the surface. We have varied this parameter performing 6 simulations with different approach distances dca , taken as a zero the jump-to-contact point. A small tip movement resulted in small clusters containig 20 atoms, with increasing the closes approach distance, the clusters grow larger around 350 atoms, Fig. 4.10 shows these effect, note that the shape of the clusters is rather irregular. Pure lead clusters were obtained in all cases, formation of any alloy was not observed, contrary to the Pd on Au(111) system studied in the previous section. This is reasonable, since lead and gold do not form bulk alloys, because they differ greatly in size (Pb is 7 % bigger than Au), and the alloy heat of solution for

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Figure 4.10: Pb clusters obtained after withdrawing the tip for 6 different simulations. Dark and light spheres denote Pb and Au respectively. Distances of closest approach dca and the number of atoms in the clusters were: a) (-2.5 Å: 20 atoms) ; b) (-3.2 Å: 46 atoms) ; c) (-5.0 Å: 132 atoms); d) (-7.6 Å: 147 atoms); e) (-8.4 Å: 350 atoms); f) (-10.5 Å: 251 atoms) a single substitutional impurity for P b/Au is +0.03eV and Au/P b is +0.01eV . Also, we can note the cohesive energies for bulk Pb is (−2.09eV ) and for Au (−3.93eV ) indicating that lead is softer than gold expected the deposition of lead from the STM tip is relatively easy, and no damage was made to the surface also for hard penetrations.

Cluster stability As mentioned above, R. Ullmann [29] has found that 3D Pb clusters can be fabricated at potentials close to the bulk deposition potential for Pb (-2.09 eV), and observed that the cluster remain stable on the surface during a very short period

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of time. As all the clusters obtained by the TIMD method in our simulations contain no gold atoms, pure lead cluster were obtained. They are less stable than bulk Pb and were complete dissolved at µ = −2.09eV by means of grand canonical Monte Carlo simulations.

In order to investigate the stability of the lead clusters in detail, the time evolution of a few examples was studied by means atom dynamics simulations. A first series of simulations were performed with small clusters, for example 20 atoms, and in all cases the pure Pb clusters were unstable with respect to disaggregation to yield 2-D structures.

Figure 4.11: Time evolution of Pb clusters; upper part, a cluster containing 20 atoms at (a1) and the final configuration after 3 ns (b1). Lower part, a cluster composed of 78 atoms (a2), an intermediate configuration at 5 ns (b2) and the final configuration after 20 ns (b3)

The upper part of figure 4.11 shows the initial configuration of a cluster containing 20 atoms obtained by the TIMD method. At the right side, the final con-

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figuration after 3 ns is shown; a cluster composed of 3 fcc(111) atomic layers transforms into a monoatomic 2-D island. This real dynamic observation reveals the great unstability of small pure Pb clusters. Simulations with larger pure Pb clusters were also performed, as an example in the lower part of figure 4.11 a cluster containing 78 atoms is shown. In frame (a2) we can see the cluster obtained by the TIMD method containing 5 atomic layers fcc (111), in frame (b2) an intermediate configuration of the cluster after 3 ns shows that the two topmost layers are not present, instead the first and second layer of the cluster have increased the diameter; this observation can be explained in terms of the high unstability of the topmost atoms because of the low coordination number. In frame (c2) the final configuration after 20 ns shows a cluster containing 2 atomic layers and a larger diameter with respect to the intermediate configuration. These observations reveals how a Pb cluster spreads on the surface even for large clusters, showing a marked unstability. It is interesting to remark that the Pb clusters present a smooth mobility, a phenomenon observed for small and larger clusters. One can speculate about the possibility of alloy formation in the experiments, and for this reason we have taken a cluster containing 78 atoms and replaced some lead atoms by gold at random positions in order to perform atom dynamic simulations. The results show clearly the great unstability of the bulk Pb/Au alloy. In figure 4.12 the time evolution of the Pb/Au cluster is shown. Frame (I) shows the system after 0.2 ns. As can be appreciate the gold atoms are surrounded by lead. Followed the dynamic for example at 1 ns (II) we can observe aggregation of Au atoms into lead, showing at the same time a decreasing height. We can note also that the cluster is rather irregular in shape, and the fcc (111) structure was lost. The final configuration is shown if frame (III) after 3 ns, these images clearly show that Pb/Au cluster are highly unstable, and therefore nanodecoration of Pb clusters on Au(111) is not possible even if alloyed clusters

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Figure 4.12: Time evolution of Pb/Au alloy cluster containing 78 atoms. I) state after 0.2 ns of equilibration, II) intermediate state at 1 ns and III) the final configuration after 3 ns

were obtained in experiments. Finally a comment about the Pb UPD monolayer. Experimentally a inconmmensurate hexagonal structure with a rotation angle of 2.5 degrees has been observed, in our simulations using the embedded atom method we have observed a rotation angle of 2.7 degrees, a result that is in good agreement with the experimental evidence. Concerning the Pb UPD monolayer there is a controversy about his composition, Toney et. al [35] have observed by means STM and AFM studies, a pure Pb monolayer. However, Green and Hanson et. al [36] have evidence of surface alloy formation on the monolayer, and they suggest a model in which the alloy formation results from lateral and vertical place exchange between Pb and Au on the vecinity of monoatomic steps. We have performed simulations of cluster generation also with Pb/Au surface alloy monolayer and found in all cases 2-D or 3-D small clusters composed only of Pb; gold was never observed in the cluster, confirming the conclusion that the experimentally observed lead cluster must be composed of pure lead.

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4.2.3 The systems Cu on Pt(111) and Ag on Pt(111) Cluster generation Nanostructuring of Cu on Pt(111) and Ag on Pt(111) are two interesting systems because they show a very large difference in cohesive energy. Denoting the cod hesive energy of the deposit (d) by Ecoh and the cohesive energy of the substrate s d s surface (s) by Ecoh , then the difference ∆Ecoh = Ecoh − Ecoh of cohesive energy

for the system Cu/Pt is ∆Ecoh = −2.23 eV, and for Ag/Pt it is ∆Ecoh = −2.92

eV. These large differences between cohesive energies mean that the deposits (Cu

and Ag in this case) are much softer than the substrate surface and therefore the transfer of material must be relativelly easy from the tip to the surface. In addition, the time of contact may be too short for place exchange to take place, therefore the formation of alloyed clusters is unlikely. Although these systems have not been investigated in experiments yet, they can help to understand the basis of cluster generation by our energetic interpretations that will be presented at the end of this chapter.

We have performed atom dynamics simulations using the same procedure described above for the generation of metal clusters with the TIMD method. Figure 4.13 shows a series of snapshots taken during the generation of Cu clusters on Pt(111). Frame (a) shows the initial configuration with the STM tip positioned at 8.9 Å from the substrate surface, a distance larger than the cutoff distance for the EAM potential. The STM tip in this case consists of a fcc(110) structure parallel to the surface with fcc(111) facets on the borders. In frame (b) the jump-to-contact point is shown, note that the displacement of material is from the tip to the surface. Frame (c) shows the closest approach distance, and finally after retraction of the tip a pure Cu cluster containing 211 atoms is shown. The obtained cluster is

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Figure 4.13: Snapshots taken during the generation of Cu clusters on Pt(111). Frame a) initial configuration, b) jump-to-contact point, c) closest approach distance and d) the final configuration with the cluster left on the surface after retraction of the STM tip. Dark and light spheres denote Cu and Pt respectively. composed of six fcc(111) layers with a truncated hexagonal pyramid shape, and 130, 36, 24, 13, 7 and 1 atoms respectively in each layer. Is interesting to remark the tendency of the first layer of the cluster near to the surface to spread on the substrate surface. This observation can be explained in terms of underpotential deposition UPD, for this system the EAM potentials predicts an ∆φU P D = 0.11 eV for a 1x1 layer. In the case of Ag on Pt(111) the difference between cohesive energy ∆E coh is larger than for Cu/Pt(111) therefore according to our previous results we expect

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bigger clusters. For example, figure 4.14 shows three stages during the generation of such clusters. In frame a) the jump-to-contact point, in which a double connecting neck was formed, due to strong interaction between Ag and Pt. In the retraction cycle, a large nanowire of around 37 Å height was observed (frame b), indicating the tendency of Ag to be deposited on Pt. Finally frame c) shows the final configuration. A cluster containg 379 atoms was left behind on the surface containing only Ag atoms. A remarkable point is that the closest approach distance for both (Cu/Pt and Ag/Pt) was the same.

Figure 4.14: Snapshots taken during the generation of Ag (dark spheres) clusters on Pt(111) (light spheres). Frame a) jump-to-contact point, b) a configuration during retraction cycle and d) the final configuration with the cluster left on the surface after breaking the connecting neck The shape of these clusters are similar to those obtained in the case of Pb on Au(111), in which the clusters were very irregular in shape and height. The great difference between the cohesive energies of both metals leads to big clusters. In most cases the clusters consist only of the deposit, this is because the simulation time (and perhaps the experimental time) is not sufficient to exchange both materials.

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Cluster stability In order to check our idea about the stability of the clusters generated by the mechanical method, we have performed grand canonical Monte Carlo simulations starting with the clusters formed by TIMD. The simulation procedure was the same as in previous systems, therefore only the results are presented. Figure 4.15 shows the configuration of the system at different chemical potentials, for example, in frame (a) the initial configuration taken from a MD simulation is shown. The equilibrium potential for Cu is -3.54 eV, and at a potential more positive (-3.465 eV) the three topmost layer of the cluster disappear and at the same time the UPD monolayer grown, the final state at µ = −3.465 eV is shown in 4.15-b). When the chemical potential is switched to -3.525 eV the cluster dissolve to only one layer, and the monolayer remains

on the surface as shown in frame (c). Finally frame (d) shows the final state at a chemical potential more negative (-3.555 eV) than the equilibrium potential for bulk Cu, and we can observe that the UPD monolayer remains on the surface as predicted by the embedded atom potential.

For the system Ag on Pt(111) we have performed also MC simulations at different chemical potentials, and the evolution of the number of particles as a function of Monte Carlo steps (MCS) is shown in figure 4.16. Due to UPD, the Ag monolayer remained on the surface at all chemical potentials considered. At µ more positive than -2.8 eV, growth of bulk Ag was observed starting from the cluster, and at µ more negative than -2.89 eV only the UPD monolayer remains stable on the surface after the complete dissolution of the pure cluster.

In summary, all Monte Carlo simulations of Cu and Ag clusters on Pt(111) showed that they are less stable than the respective bulk metals. They were com-

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Figure 4.15: Configurations at the end of the grand canonical MC simulations for Cu on Pt(111). a) Initial configuration, b) at µ=-3.465 eV, c) at µ=-3.525 eV and d) at µ=-3.555 eV pletely dissolved at the correspondig bulk binding energies (µ Cu =-3.54 eV and µAg =-2.85 eV).

4.2.4 The system Cu on Cu-island on Ag(111) In the systems studied originally by Kolb and co-workers, they mentioned that it is difficult to generate Cu clusters on Cu(111) under the experimental situation. Then for this problem they chose an alternative way in which Cu clusters were generated on a Ag(111) surface with an island of 2 layers height of Cu [26, 27]. Based on this experimental situation, we have performed Atom Dynamic simulations also for this particular system. Figure 4.17 shows a typical series of snapshots taken during the formation proccess. Frame a) show the initial state, b) jump

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1000 µ = −2.75 eV µ = −2.78 eV µ = −2.89 eV µ = −2.92 eV µ = −2.935 eV

N

800

600

400

200

0

1000

2000

3000 4000 MC steps

5000

6000

7000

Figure 4.16: Evolution of the number of particles as a function of MCS for a Ag cluster on Pt(111) at different chemical potentials

to contact, d) closer approach distance, and from d) to j) different stages during the retraction cycle. In the initial configuration the Cu island of square shape with 22.3 Å of side is composed of 170 atoms. At the end of the Molecular Dynamic simulation a cluster is left behind on the surface, such cluster has a radius of approximately 7.6 Å and contains 97 atoms indicating that the height of the cluster is bigger, but the transference of the material was from the surface to the tip, contrary to the Cu/Cu(111) system that we will see in the next chapter. We can note here, that the difference in the cohesive energy between Cu tip-atoms / Cu islandatoms is zero of course, however the difference between cohesive energies Cu-Ag is +0.69 eV and Cu-Pt −2.23 eV, indicating that the atoms on the island are easier

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to pull up than the atoms on the tip. These simulations show that it is not possible to deposit Cu on Cu-island on Ag(111) as mentioned by Kolb in [26, 27]. Only detachment of atoms from the island is possible, in other words, it is as if we reorganized the island into another configuration.

Figure 4.17: Formation mechanism of a Cu cluster generated onto a Cu-island supported on Ag(111) When the identation of the tip is larger, we have observed that the Cu atoms in the island are completely detached and a hole is left behind on the Ag(111) surface, (as expected because of the difference in cohesive energy mentioned above).

4.2.5 Review of systems studied This part of the thesis is a project in collaboration with M.G. Del Popolo and E.P. Leiva from the University of Cordoba (Argentina) as mentioned above. Then, a review of the systems studied by Del Popolo and Leiva are given as a complementary information of great interest to our work.

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The system Cu on Au(111) The generation of Cu clusters on Au(111) was the first system experimentally studied and subsequently employed as a prototype for the development and improvement of the deposition technique [25, 26, 34]. The clusters proved to be stable up to potentials of about 200 mV positive of the dissolution potential for bulk copper. This was quite unexpected, since clusters are normally less stable than bulk material because of their unfavorable surface-to-volume ratio.

Figure 4.18: Typical Cu-Au clusters generated by the STM tip. Upper figures correspond to lateral views, lower figures correspond to top-views of the surface. The closest approach distances were a) -4.8 Å b) -6.6 Å c) -8.4 Å . The simulations carried out by Del Popolo [30, 31] were performed for a Au(111) substrate covered with a Cu monolayer, using three types of STM tips ([111], [110] and amorphous). Gold-rich clusters were obtained in all cases with [111]-type tip, with the generation of surface damage even for moderate penetration of the tip. Typical clusters resulting from these simulations are shown in

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figure 4.18. On the other hand, [110] and amorphous-type tips generated larger pure Cu clusters, the damage being confined to the Cu monolayer. Grand canonical Monte Carlo simulations were also performed by Del Popolo, and the results show that gold rich clusters are more stable than pure ones; this is in agreement with the results presented for Pd clusters on Au(111). The system Cu on Ag(111) Experiments have shown that is not possible to generate 3-D copper clusters on Ag(111) [28], since the approach of a Cu loaded tip to the surface just produces some disperse 2-D islands that yield no reproducible patterns. For this system a set of simulations was performed by Del Popolo [31]. A set of six simulations with different closest approach distance dca are shown in figure 4.19. Just holes of different depths were dug into the surface. Simulations with [110]-type tips were also performed and very small clusters were obtained for different closest approach distances.

4.3 A few conclusion on cluster generation with a STM tip Below we summarize what we have learned about the formation of clusters and their stability. Cluster formation The formation process consists of three stages: 1. The approach of the tip to the surface until the jump-to-contact. This latter process involves the displacement of tip material to the substrate or vice

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Figure 4.19: Final configurations after different attempts of Cu nanostructuring on a Ag(111) surface with a A-type tip. dca are a) -1.2 Å , b) -2.4 Å , c) -3.6 Å , d) -4.8 Å , e) -6.0 Å , f) -9.0 Å . versa. Without consideration of tip geometry, the metal with the lower binding energy jumps towards the other. 2. Indentation of the substrate surface. This is a strong mechanical process that takes place on the contact region. 3. Retraction of the STM tip, elongation and finally rupture of the connecting neck. Cluster size and composition In general terms, the size of the clusters increase with the distance of closest approach of the tip to the surface. In those cases where alloying was possible, the clusters became progressively enriched with the material of the substrate surface.

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Conditions for the formation of stable clusters We can now summarize our results and formulate the conditions for the generation of stable clusters with the electrochemical STM. For the formation of the clusters, the most important parameter is the difference between the cohesive energy of the deposit d and of the substrate s: d s ∆Ecoh = Ecoh − Ecoh . Soft tips tend to form large clusters, so in this respect

a large negative value of ∆Ecoh is favorable. Conversely, hard tips dig holes into soft substrates, or at best generate an island composed of substrate atoms. For the clusters to be useful they must be stable over a certain range above the dissolution potential for the bulk deposit – they cannot be kept below the dissolution potential or else they will grow. Thermodynamically they may be stable if they form an alloy with the substrate. Therefore the alloy heats of solution for single substitutional impurities ∆Hd are decisive for the stability of the clusters. However, when the deposit is much softer than the substrate, there is practically no mechanical mixing of the two types of atoms. In addition the time of contact may be too short for place exchange to take place. Therefore, the optimum conditions for the formation of stable clusters are: (1) a deposit that is a little softer than the substrate; (2) the formation of a stable alloy between the two metals. In Table 4.1 we have summarized data for all systems that have been investigated either experimentally or by simulations. Four types of systems can be distinguished: (1) Systems with negative heats of alloy formation and negative values of ∆Ecoh (deposit softer than substrate). These form stable alloyed clusters. (2a) Systems that form surface, but not bulk alloys, and with negative values of ∆Ecoh . The systems that have been investigated involve Pb atoms, which are too large to form stable alloys with Au or Ag. These may form unstable clusters. (2b) Systems with a very large negative values of ∆Ecoh (deposit much softer

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System

Experiment

Simulation

∆Ecoh /eV

∆Hd /eV

Pd on Au(111)

3D-stable

3D-alloying

-0.02

-0.36, -0.20

Cu on Au(111)

3D-stable

3D-alloying

-0.38

-0.13, -0.19

Ag on Au(111)

3D-stable

3D-alloying

-1.08

-0.16, -0.19

Cu on Pd(111)

-

3D-alloying

-0.37

-0.39, -0.44

Pb on Ag(111)

3D-unstable

-

-0.76

-

Pb on Au(111)

3D-unstable

3D-pure

-1.84

0.03, 0.01

Cu on Pt(111)

-

3D-pure

-2.23

-

Ag on Pt(111)

-

3D-pure

-2.92

-

Cu on Ag(111)

2D

hole, 3D small

0.69

+0.25, +0.39

Ni on Au(111)

hole

hole

0.52

+0.22, +0.28

Co on Au(111)

2D gold

-

0.48

0.4, -

Rh on Au(111)

holes

-

0.52

0.22, 0.28

Table 4.1: Classification of experimental and simulated systems according to their difference in binding energy ∆Ecoh , and alloy heats of solution ∆Hd for single substitutional impurities. The experimental results can be found in ref. [29, 34, 25, 26, 27, 28], the information corresponding to Ni-Au(111) simulations was taken from [37, 38]. ∆Ecoh and ∆Hd were taken from [5]. than substrate). For example Cu and Ag on Pt(111). These may form pure unstable clusters (4) Systems that do not form alloys; the systems that have been investigated all have tips that are harder than the substrates. No, or only unstable, clusters have been observed. 3D-stable means that the clusters generated in the experiments endure dissolution. 3D-unstable means that experimental clusters dissolve easily. 2D means

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that only 2D islands were obtained. "Hole" refers to the fact that only holes were generated on the surface after the approach of the tip. Concerning the column of simulation, "alloying" means that the cluster contained not only atoms of the metal being deposited but also some coming from the substrate.

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Chapter 5 Consequences of Tip-Surface interactions 5.1 Introduction In previous chapter the generation of metal clusters with the so-called tip induced metal deposition method was shown. In these studies, we have analyzed different metal couples, and arrived on the conclusion that optimal systems for nanostructuring are those where the metals participating have similar cohesive energies and negative heats of alloy formation. An interesting issue still deserving further study is the effect of the cristalline nature of the tip on the nanostructuring process. For example, some interesting questions concerning cluster generation on a surface of the same nature as that of the metal being deposited are: • Is it possible to transfer metal from a tip onto a substrate of the same material ?

• Is there a jump to contact between two metals of the same kind ? In which direction?

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• Do different tip geometries lead to remarkable differences in cluster formation ?

We have performed extensive Atom Dynamic simulations (AD) in order to answer these questions. We present first the results concerning Cu cluster generation on Cu surfaces with the Tip induced metal deposition method using in this case different geometrical STM tips, that we call [111] and [110] depending of the type of surface facing the surface of the substrate.

5.2 Calculation Method The simulation method employed in the present work is similar to that used in previous chapter. Concerning the initial configuration used here, our model system consists of a semi-ellipsoidal tip of fixed platinum atoms from which several layers of mobile copper atoms are suspended. The electrode is modeled by eight layers of copper atoms arranged in fcc(111) geometry; the top six layers are mobile and the rest fixed in order to mimic the bulk metal. The interatomic potential used to compute the interaction between the atoms is the Embedded atom method (EAM) [5]. The semiellipsoidal tip was obtained by sectioning a piece of Pt fcc bulk structure, that was oriented with a (111) or a (110) single crystal face facing the surface of the substrate. This oriented the suspended Cu atoms in the same direction. In the following, we denote the two types of tips with [111] and [110] respectively. The tip was then moved at a constant speed of 3x109 Å s−1 towards the surface; after each step the system was allowed to equilibrate for 20 ps. The ”jump to contact” process, in which a bridge between the copper atoms on the tip and the substrate is observed, served as our reference point, from which the tip was moved

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a preset dca distance further. The tip was subsequently withdrawn with the same speed, a connective neck formed, which broke and left a cluster behind on the surface with both types of tips.

5.3 Influence of tip structure in homoatomic systems Figure 5.1 shows the two types of STM tips mentioned above. As we can see in Figure 5.1 left) the crystallographic (110) planes are located parallel to the surface of the substrate and (111) planes are apparent at the curved part of the tip.

Figure 5.1: STM type tips employed in the present simulations. (Left) type [110] and (right) type [111]. The (111) and (110) facets are shown by lines

In the case of [111] tips (Figure 5.1 right) the planes parallel to the surface are [111], and the boundaries present no clear orientation. Cluster generation can be described in terms of three stages: I) motion of the STM tip towards the surface, II) jump to contact process and III) withdrawal of the tip, with the concomitant appearance of a cluster on the surface.

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5.3.1 Tip-sample approach This process involves displacement of the tip towards the surface and the formation of the jump-to-contact (JTC) which is accompanied by a marked decrease in the potential energy of the system.

Figure 5.2: Initial ((a) and (c) ) and final ( (b) and (d) ) configurations of typical systems. a) and b) correspond to [110] tips; c) and d) are [111] tips. Atoms marked in gray are those transferred from the tip to the surface. Atoms marked in white are those remaing on the tip. Atoms marked in black are those belonging to the surface of the substrate. Black circles represent also Pt atoms, fixed on the tip during the simulation. This process has different characteristics for the two types of tips employed here. In the case of the [111] tip, the atoms jumping to contact belong to the first (111) plane of the tip facing the surface, which becomes partially destroyed.

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This yields at the beginning a rather asymmetric contact point. On the other hand, the jump-to-contact process in the [110] tips involves atoms arising from the first 4-5 (110) lattice planes of the tip facing the surface. In this proces, only atoms belonging to the outer part of the tip parcipate, through the diplacement along the (111) facets of the tip. The relatively easy gliding along the (111) facets is the main reason for the transfer of atoms from the tip to the surface without surface damage of the latter.

5.3.2 Tip-Sample contact Indentation: This may occur upon further displacement of the tip towards the surface. Here the behavior of the two types of tips is also markedly different. In the case of the [111] tip, the approach produces the indentation of atoms belonging to the tip into the substrate, and reciprocally, some atoms are expelled from the surface, so that a mixing results. In the closest approach conditions used here (dca = −5.32 Å), five atoms penetrated beyond the surface plane. This behavior

must be contrasted with that of the [110] tip, where no indentation of the atoms from the tip was produced for the simulation conditions employed here.( −5.5

Å < dca < 0). This effect can be observed in figure 5.2 in which the atomic

coordination of both tips at the beginning and at the end of the simulation have been plotted. At this stage, it is interesting to monitor the behavior of the [110] tip at the distance of closest approach dca of −0.108 Å . On one hand, the substrate has the (111) orientation. On the other side, the tip exhibits the (110) orientation. The question is, how the transition from one orientation to the other at the connecting neck takes place. Figure 5.3 shows the structure of the three first layers of the tip in contact with the surface. It can be observed that while the first layer has clearly the (111) orientation and the third layer has a (110) structure, the second layer

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2

3

4

5

6

7

8

N













×

N













×

×

N











×

×

×

N

N







×

×

×

N

N







×

×

×

×

N







×

×

×

×

N







×

Table 5.1: Layer structure for different number of layer, counting starts from the closest layer to the tip at different distances during a retraction cycle. The closest approach distance was dca = −0.108 Å. (×) (111), ()(110), (N)mixed shows a transition behavior.

Figure 5.3: Structure of the three first layers of the [110] tip in contact with the surface during the nanostructuring process at dca = −0.108 Å. This corresponds to the closest approach distance in the simulation shown here.

When retraction of the tip occurs, more(110) layers are turned in the (111) orientation, as shown in Table 5.1. The results shown in this table are typical, in the sense that at the most 4 atomic layers at the bridge between the tip and the substrate are oriented by the substrate.

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5.3.3 Retraction After the closest approach distance the tip is retracted. This proccess produces a connecting neck that is elongated and finally breaks. This stage is also strongly different for both tips. While the retreat of the [111] tips produces an important conical shaped strain field in the substrate that is evident by visual observation of atomic positions in the simulation, this is not observed in the case of the [110] tips. This is shown in figure 5.4

Figure 5.4: Geometric configuration of a [110] tip(left) and of a [111]tip (right) during elastic deformation in the retraction cycle . Note the conical strain field generated on the substrate in the case of the [111] tips. The variation of the component of the force on the tip perpendicular to the surface of the substrate F z reflects the different behavior of the two types of tips. In Fig. 5.5, it can be appreciated that at the JTC point F z decreases abruptly for both tips, following a stronger attraction in the case of the [111] tips. Further approach produces a strong repulsion in the case of the [111] tip, that has also a strong attractive counterpart when the tip is retracted. Further retraction of the [111] tips shows a marked hysteresis on F z due to the rearrangement of the atoms in the

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connecting neck, presenting a series of elastic and plastics changes. While the descending part of the forces in the retraction cycle coincide with observation of the strain cones mentioned above, the ascending portion corresponds to the relaxation of the system, where stress is released. The behavior of the [110] tip is markedly different, the histeresys in the force is considerably reduced, without any structure noticeable in the retreat cycle. Note that the |dca | values in these simulations were

kept relatively low, so that no plastic relaxation could be observed in the forces during penetration of the tip.

2 [111] Tip [110] Tip

1

Fz (nN)

0

-1

-2

-3

-4 -4

-2

0

2

4

6 d_ca [Å]

8

10

12

14

16

Figure 5.5: z component of the force on the tip F z as a function of the tip-substrate distance. full line: [110] tip, dotted line [111] tip. The well defined behavior of the [111] tips allows a more quantitative evaluation of properties. In this case, we attempted the calculation of the Young’s modulus from the force exerted on the tip and the displacement of the lattice planes during the retraction cycle. With this purpose, the average distance < d 12 > between the first and the second plane of the tip was calculated, and the Young’s

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modulus Y was obtained from the force on the tip F according to the relatioship:

Y =

F A (< d12 > −d111 )

where d111 is the distance between (111) lattice planes in the absence of external forces. The area A was estimated from the average number of atoms in the first and the second lattice plane, and the atomic surface of a Cu atom on the (111) lattice plane. The force on the tip was plotted as a function of < d12 > −d111 and Y was calculated from the slope. We obtained a value of 200±10 GPa . This can

be compared with the experimental value for the (111) direction of 192 GPa [40] or with that calculated from the experimental elastic constants of reference [41]. The EAM value for the (111) direction is Y =188 GPa. Figure 5.6 shows the final configuration of the surface after withdrawn the STM tip for different tip approaches with both tips [110] a) and [111] b). It can be generally stated that the closer the tip approaches the surface, the larger the generated clusters. It can also be appreciated that a hole is left at the right side of the cluster for dca < −3Å in the case of the [111] tips. On the other hand, surface damage is never observed with [110] tips.

In order to analyze which atoms of the tip are involved in the formation of the cluster, it is instructive to draw the configuration of the tip before the contact, marking those atoms that are finally transferred to the surface of the substrate. This has been made in figure 5.2. The formation of clusters with [110] tips involves the transfer of a whole piece of the tip, becoming more evident compact (111) facets on it after the transfer. On the other hand, it is clear in the case of [111] tips that the transfer of atoms is less homogeneous. Even atoms of the tip close to the surface are not transfered. Table 5.2 reports for each layer of tip atoms, starting from that close to the surface, the total number of atoms and

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Figure 5.6: Final configuration of the surface after the nanostructuring proces with [110] tips (a) and [111] tips (b) for different tip approaches. The closest aproach distances dca were, increasing from left to right. the number of atoms transfered to the surface after the simulation. Reciprocally, while no atom transfer takes place from the surface to the tip in the case of the system[110], some atoms do transfer from the surface to the [111] tip, a fact that can be appreciated in figure 5.2d)

5.4 Comparison with the behaviour of the forces in heteroatomic systems Depending on the structure of the tip, we have seen strong qualitative and quantitative differences in the behavior of the forces on the tip upon retraction in the case of the present homoatomic system. One could wonder if this is just due to a problem of commensurability between tip and substrate. For example, simulation of a (001) MgO [42] contact shows that a rotation of 45 o of the probe leads to jumps in the force that are smaller in magnitude and much less abrupt than in

5.4 C OMPARISON

WITH THE BEHAVIOUR OF THE FORCES IN HETEROATOMIC

79

SYSTEMS

Layer # NT [110] Ntrans [110] NT [111] Ntrans [111] 1

3

3

16

9

2

10

10

67

46

3

18

18

103

41

4

27

27

127

11

5

32

32

156

4

6

41

41

171

0

7

48

47

188

0

8

52

34

205

0

Table 5.2: Total number of atoms in each layer NT of the [110] and [111] tips shown in figure 5.2a) and c) number of atoms of each layer transfered N trans to the surface after the simulation. The layer number are counted starting from the lattice plane closer to the surface

the commensurate systems contact. For this reason it is interesting to compare the forces of the homoatomic system (Figure 5.5) with those of hetheroatomic systems with different atomic sizes, where commensurability effects should be absent. This is for example the case of Cu nanostructuring on Au, which delivers the forces shown in Figure 5.8. Although this system is in principle incommensurate because of the different atomic sizes, the same qualitative features appear as in the homoatomic system. Figure 5.7 shows results of the forces for the Pd/ Au(111) system, confirming the general trend. As stated before, it is the relatively easy gliding of the (111) lattice planes that allows the transfer of matter in the (110) tips withouth the appearance of strong strain fields in the substrate.

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20 Tip [110] Tip [111]

Fz [nN]

0

-20

-40

-60

-5

0

5 d_ca [Å]

10

15

20

Figure 5.7: z component of the force on the tip F z as a function of the tip-substrate distance for nanostructuring of Pd on a Au(111) surface, full line: [110] tip, dotted line [111] tip.

5.5 A few conclusions on STM characterization By performing atom dynamics simulations for the deposition of nanosized metal clusters on (111) single crystal electrode surfaces of the same material by means of an STM tip, and keeping the penetration depth of the tip relatively low, we showed that the crystalline orientation of the tip plays an important role in this process.

When the tip orientation is such that a (111) facet of the tip is facing the surface, the nanostructuring process is accompanied by the development of a stronger perturbation of the surface than that observed for more open structures on the surface of the tip. This can be appreciated both in the strain field generated in the substrate and in the damage after the nanostructuring process. More importantly, this is also manifested in a measurable property: the force on the tip. When the

5.5 A

FEW CONCLUSIONS ON

STM

CHARACTERIZATION

Tip [111] Tip [110]

0

Fz [nN]

81

-10

-20

-30

-2

0

2

4

6 d_ca [Å]

8

10

12

14

Figure 5.8: z component of the force on the tip F z as a function of the tip-substrate distance for nanostructuring of Cu on a Au(111) surface, full line: [110] tip, dotted line [111] tip.

compact (111) face is facing the substrate, the force shows a well defined sequence of elastic and plastic changes, which are absent in the case of more open faces, like the (110). Thus, an exciting potential technological application appears: the possibility of orienting a nanocristallite deposited on a tip. If the relative orientation of the tip would be varied with respect to a (111) surface of a substrate, the occurrence of well defined changes on the force on the tip perpendicular to the surface would be an indication that the cristallite on its top is being oriented with the (111) parallel to the surface. The experiments should be done applying a "soft touch" to the surface by the tip (no important plastic deformation), that is enough to set up a strong interaction between the tip and the surface when both are faced showing the (111) plane. This strong interaction is also expected to occur for heteroatomic systems, as we found for Cu(111)(tip) / Au(111)(substrate), Pd(111)(tip) / Au(111)(substrate), Ag(111)(tip) / Au(111)(substrate), so that com-

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mensurability between adsorbate and substrate is not a requirement.

Chapter 6 Generation of Metal Nanowires 6.1 Introduction As the scale of microelectronic engineering continues to decrease, the nature of electron transport throught low-dimensional nanometer-scale channels such quantum wires [11] and carbon nanotubes [12] has become important. Metal nanowires are one of the most attractive materials because of their properties and applicability in future technologies. Metal nanowires are also attractive because they can be fabricated with various techniques [13], and are the center of attention of many research groups at the present. The dimension of such nanowires range from single one-dimensional chains of atoms to a few hundreds of atoms in diameters, but in lengths they can take from a few atoms to many microns. Because this great range in the aspect ratio d/l, different names have been used in the literature to describe the wires in order to reflect the differents shapes. For example wires with small aspect ratio than 20 are called "nanorods", while aspect ratio > 20 determined the so-called "nanowires", also when short wires are bridged between two surfaces, they are often referred to as "nanocontacs". As mentioned above electron transport prop-

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erties are important, and metal wires are also decribed in term of classical and quantum wires. The classical behaviour arises because the wires are much longer than the electron mean-free path, and much thicker than the electron Fermi wavelength. In the second case "quantum" wires, the mean-free path is a few tens of nm and the electron wavelength is only a fraction of nm, comparable to the size of an atom [13]. When the wire is shorter than the electron mean-free path, the electron transport is ballistic, that is, without collisions along the wire. If, in addition, the diameter is comparable to the electron wavelength, quantum-size confinement becomes important in the transverse direction. The conductance G of the system measured between two bulk electrodes is described by: G=

2e2 N h

(6.1)

where e is the electron charge, h is Planck’s constant, and N is the total number of modes with unit transmission probability. In this chapter, we will see the fabrication of quantum nanowires through mechanical and electrochemical methods, and we shall discuss some geometrical and thermodynamic properties of their stability.

6.2 Mechanical Method Mechanical methods generate a metal wire by mechanically separating two electrodes in contact. During the separation process, a metal neck is formed between the electrodes due to strong metallic cohesive energy, which is stretched into an atomically thin wire before breaking. One of the most used method to produce this wires is with the aim of a STM tip [11] as one electrode and a single crystal surface as the other one.

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We have adapted the experimental situation to atomic dynamic simulation. All the simulations were made with the atom dynamic code "XMD" developed by J.Rifkin [39]. We have used a configuration similar to that used in our previous simulations of tip induced metal deposition, in which a Au (111) tip covered by Cu is brougth into contact with a electrode surface, in this case of Au(111) covered with the experimental predicted Cu monolayers. For the atomic interaction, we have used the EAM method described in chapter 3. After the maximal approach distance the tip is retracted in steps of 0.006 Å every 2 ps, resulting in an average velocity of 30 cm/s; figure 6.1 shows a series of snapshots taken during the retraction of the tip.

Figure 6.1: Snapshots taken during the elongation process, the arrows show glide planes. Dark and light spheres represent Cu and Au atoms respectively. In frame a) we can see the maximal approach of the tip to the surface (initial

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configuration), subsequently frames b-c-d-e and f) shows the retraction process, frame b) shows a well ordered fcc (111) structure on the connecting neck, however between frame b) and c) we can observe an abrupt structural transformation, in which different crystalline regions with glide planes can be observed, as also from frame c) to d), but in this case the glide planes have been changed. The structural transformations lead to an elongation of the wire as we can see in frame f). In the narrow constriction we have observed steps in which the planes lose their fcc (111) orientation, showing a marked disorder in the narrow layers. Figure 6.2 shows this effect, in which we have plotted the atomic density from the topmost layer of the substrate surface to the last planes of the tip as a function of the distance perpendicular to the surface. The dashed line represents a forward step, in which this disorder appears between 15-20 Å . Accompanying these processes are variations in the cross sectional areas and shapes of the wire. We have ploted in figure 6.3 each cross-sectional area for each layer of the last configuration shown in figure 6.1. All layers present a quasi-perfect fcc (111) structure, except for the topmost layers (15-19) in which a marked region can be observed on the left-top part of the layer. Also we can appreciate that the area, and the shape is different from layer to layer, but in all cases with the minimal boundary possible. Layers 1 to 3, shows a certain amount of Au atoms, that were pulled out from the subtrate surface, this effect is also seen in figure 6.2. To gain insight on the mechanical behaviour of the structural transformation between various steps, the z-component of the force Fz (along the wire direction) has been computed. Fz was recorded during elongation of the wire and is shown in figure 6.4. The oscillatory pattern of the forces are in concordance with our previous results shown in chapter 4. We can observe hard linear jumps on the force at the first stage of the elongation, because of structural transformation in

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Figure 6.2: Atomic density as a function of the z-direction for two different retraction points the neck. Following the elongation a wire starts to form, with successive jumps in the force. In most cases, a rearrangement also induces a sharper decrease on the smallest cross-sectional area of the contact (perpendicular to the z−axis). There are, however, structural transformations that do not affect the crosssectional area at the thinnest point of the nanowire. We can note also, that some force jumps are more abrupt than others, in other words the slope of the jumps are differents, indicating a compromise between tension and atomic rearrangement. As mentioned above the smallest cross-sectional area S was also monitored, this quantity is calculated by the procedure used by Bratkovsky [14] and is expressed in units of number of atoms. S is approximately equal to the Sharvin conductance

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Figure 6.3: Cross-sectional areas of the wire, from layer 1 to 19. on the nanocontact in units of G0 =

2e2 h

given by [15]:

Gs = G0 πS

(6.2)

Figure 6.5 shows S curve during elongation of the wire. The jumps in S are correlated with the jumps of the force as mentioned above.

6.3 Electrochemical Method 6.3.1 Introduction - Experimental Background Li and Tao demostrated an electrochemical method to fabricate metal quantum nanowires using an electrochemical STM setup [11]. In this method they posi-

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Figure 6.4: Variation of the force with the elongation along the z−axis for a Cu nanowire tioned a STM tip at a fixed distance of (10-150 nm) from a Au substrate electrode in a plating solution (Cu). Copper is then deposited onto the sharp STM tip. The selective plating onto the tip is achieved by keeping the substrate potential slightly more positive than the bulk deposition potential. When the growing Cu reaches the substrate, a contact is formed between the STM tip and the substrate surface, which is reflected in the experiment by a sharp increase in the current that flows between the tip and the surface. By slowly dissolving away the deposited Cu contact, a Cu quantum wire is formed between the tip and the substrate electrodes. The conductance measure of such wire, changes in a stepwise fashion with a preference to occur near integer multiples of G0 . As we can observe the STM tip is not used for a mechanical stretching as in the previous method, in this case is only used as a metal electrode, in which we can control the distance between both surfaces.

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140 120

S

100 80 60 40 20 0

0

5

10

15 Elongation [Å]

20

25

30

Figure 6.5: Variation of the minimal cross-sectional area with the elongation along the z−axis for a Cu nanowire

With these experiments in mind we have adapted the experimental situation in order to perform computer simulations of electrochemical deposition and etching between a STM tip and a substrate surface. In this case, we have taken a Au tip, by sectioning a piece of Au(111), and a substrate electrode composed of six layers arranged in fcc(111) orientation. The interaction between the atoms was again taken into account with the embedded atom method mentioned in previous sections. The simulation technique in this case was the Monte Carlo method in the grand canonical ensemble, in which the chemical potential µ can be controlled, by means varying of the number of particles N , at volume V and temperature T constant.

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6.3.2 Atomistic Simulation - grand canonical Monte Carlo In the experimental situation presented above, the electrode potential is controlled during deposition or etching of Cu, in other words the electrochemical potential of ions is really controlled. Under equilibrium conditions µ is equal to the chemical potential of the deposited metal. For this reason we have chosen the grand canonical ensemble, in which we can control the chemical potential of the metal of interest.

Cu nanowires suspended between a fcc (111) STM tip and a fcc (111) flat surface The initial configuration was a small Au tip and a square substrate surface with X − Y periodic boundary conditions, both with fcc [111] structure. The number

of Au tip atoms was 405 arranged in five layers, and the substrate surface was composed of three layers with 1197 atoms. The distance between the tip and the

surface was fixed at 42.3 Å. All simulations presented here were performed at room temperature (300 K). In order to gain insight into the electrochemical properties and the dependence of Cu nanowires structure on the chemical potential µ, we have started the simulation at µ=-3.35 eV, in which we fill all available sites. After this, the chemical potential was switch lightly until µ=-3.50 eV whereupon after 10000 Monte Carlo steps (MCS) a thin nanowire of 7.34 Å of minimal cross sectional area and 1148 Cu atoms was created. Figure 6.6 shows the evolution of the number of particles as a function of MCS for the nanowire obtained previously at three different chemical potentials. The nanowire grew when the chemical potential was higher than -3.52 eV, and dissolve completely when µ was smaller as show figure 6.7. We can compare this value

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Figure 6.6: Evolution of the number of particles as a function of MCS for a Cu nanowire containing 1148 atoms at three chemical potentials: µ=-3.45 eV, µ=3.40 eV and µ=-3.35 eV. The inset at the upper left shows the fluctuation of the number of atoms at µ=-3.45 eV in a enlarged scale. with the cohesive energy calculated for bulk copper with EAM, (µ Cu =-3.61 eV). The fcc [111] structure remains well defined after deposition and growth of the nanowire, and by close inspection in the growth mechanism, we can see that the atoms are first deposited in the region near to the surface, and in some cases, as for example at -3.40 eV, the minimal cross section area remains more o less stable with small fluctuations in the number of atoms and shape.

The final configurations after 10000 MCS are shown in Figure 6.8 for different chemical potentials. As we can see in this figure, at a chemical potential higher than -3.5 eV the nanowire grows and dissolves when µ is lower. As the frame a)

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-3.65 eV -3.60 eV -3.55 eV

1000

Natoms

800

600

400

200

0

0

200

400

MCS (x10)

600

800

1000

Figure 6.7: Evolution of the number of particles as a function of MCS for different chemical potentials: µ=-3.55 eV, µ=-3.60 eV and µ=-3.65 eV. shows the final configuration at -3.60 eV, as we can observe the wire was dissolved and at the same time the Cu monolayer on the surface also starts to dissolve. It is interesting to analyze the growth mechanism at µ=-3.35 eV. New ad-atoms are added at kink sites at the early stages because of the high coordination number, As we can see in figure 6.8-c the wire grew mainly in the region close to the tip and the substrate surface. Inspection of the atomic configuration reveals different planes, similar to the "glide planes" observed in the mechanical method, as was shown in the previous section. These planes are a combination of closed-packed and more open faces, in other words, a combination of [111] and [100] faces as can be appreciated in figure 6.9. Figure 6.9 shows the wire perpendicular to the X − Z plane along the Y -

direction (a), we can distinguish three regions with planes (100) and (111). Dif-

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Figure 6.8: Final configuration after 10000 MCS, a) µ=-3.60 eV, b) µ=-3.50 eV and c) µ=-3.35 eV ferent domains can also be appreciates, mostly (111) and (100) faces and some disordered regions originated mainly by the junction of different crystaline faces, also is shown the opposite Y -direction (b) in which only (111) planes can be observed on the surface of the nanowire, we can note on the right side a disorderly face.

Cu nanowires suspended between two fcc (100) flat surfaces Tosatti and co-workers [23] show in a theoretical paper that new noncrystalline nanowires, which they call "weird", should in fact be rather readly realized in nature. They have performed molecular dynamics - simulated annealing calculations using the "glue model" for the interaction between the atoms (similar in spirit to EAM), and found structures that differ from the crystalline bulk. Helical structures were predicted for Pb and Al nanowires starting with fcc (110) and (100) orientations. An interesting observation was that some wires were formed by a central string surrounded by two or three succesive coaxial cylindrical shells. After these predictions an experimental work by Kondo and Takayanagi [24] shows for first time, that fcc (110) nanowires are formed by helical multi-shell structures. Using transmission electron microscopy (TEM). The authors studied

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Figure 6.9: Nanowire generated at µ=-3.35 eV. The 3D system is projected on the X − Z plane a) Y -direction view and b) opposite Y direction view. The arrows indicate different planes or faces.

atomically thin, several nanometers long, very regular gold nanowires formed by thinning down a narrow bridge between two oriented tips. Shrunken to radii of 2 to 6 Å , these wires display "magic" preferred radii and geometries. This means that they only come in a few discrete sizes and shapes. What is more striking is that the internal structure of these magic nanowires, is weirdly noncrystalline. The structure consists of coaxial cylindrical tubes, or shells. A shell s contains a well-defined number ns of strands, which increases with the shell radius. We have performed grand canonical Monte Carlo simulation of Cu nanowires suspended between two fcc (100) surfaces. Figure 6.10 -a) shows the initial configuration of the nanowire, note that the substrate surfaces below and above the wire are not showed. The nanowire was made by cutting a bulk [100] crystal structure in a cylindrical shape. The resulting nanowire has a diameter of 5.7 Å and 34.3 Å long. We

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Figure 6.10: Initial and final configuration after 10000 MCS for a Cu(100) nanowire, a) initial configuration, b) µ=-3.38 eV, c) µ=-3.36 eV and d) µ=-3.34 eV

have performed simulations at different chemical potential in order to observe the growth mechanism of [100] wires. Figure 6.10 shows the final configuration after 10000 MCS; in frame b) the wire was exposed at a chemical potential of -3.38 eV, the diameter in the central region of the wire remains constant, the atoms are added close to the surfaces, because more kink sites are found at the contact point between the nanowire and the surface. The new atoms form fcc (111) planes around the wire, as can be appreciated from figure 6.10-b. Switching the chemical potential lightly more positive to -3.36 eV we found that the wire starts to grown also from the vicinity of the surface, and

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after a few MCS a widening of the wire along the z-direction is observed. Analysis of the atomic configuration shows that at determined MCS the wire growth in a helical fashion, as shown in figure 6.10-c. The black points on the plot trying to help the visualization . If we analyze the "matrix structure" of the fcc(100) wire in the initial configuration, from which the new wire can grow, we observe also these rolled lines, from which we can infer that they are used as a template. Closer inspection on the atomic configuration when the wire grows with this helical shell, reveals a high coordination of the atoms if they grow in the direction marked with black points in figure 6.10. In the last frame of figure 6.10 the final configuration after 10000 MCS at -3,34 eV is shown, a helical shell form in the growth mode is observed, but in this case the strands run from the surfaces.

500

suspended unsuspended

400

Natoms

300

200

100

0 -3,4

-3,38

-3,34 -3,36 Chemical Potential (eV)

-3,32

-3,3

Figure 6.11: Number of particles vs µ for suspended and unsuspended nanowires.

We have also performed simulations of unsupported copper nanowires. For this case, we have deleted the surface atoms, and kept the nanowire with the same initial configuration that for supported ones. The results are surprising because we have performed grand canonical MC calculations at the same chemical poten-

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tials and the behaviour was completely different. Figure 6.11 shows the number of atoms added or deleted as a function of MCS for both wires, supported and unsupported. The plot is clear, an unsuspended nanowire does not grow at these chemical potentials in which a suspended nanowire between two surfaces can grown. The effect of the surface is crucial because it provides the first sites for growth with high coordination numbers, which are no present in the case of a free nanowire.

Fabrication of a molecular gap between two electrodes. A novel application of technological research consist in the fabrication of electrodes with a separation of a few nanometers (nm) with the purpose of connecting one or more molecules between them [16, 17, 18]. One of the methods devised is based on the mechanical rupture of a nanowire, following the formation of two electrodes [20]. Among the non-mechanical methods are electromigration and the electrochemical method [18, 19]. Electrochemical methods involve metal deposition and dissolution as shown above. For example, in figure 6.8-a) a Cu nanowire was dissolved by switching the chemical potential close to the cohesive energy of bulk Cu. This is the case in the experiments of Tao and co-workers [13, 18, 19], who have measured the tunneling current across a nanogap using scanning tunneling microscopy, and then translated the measured values of the current into distances using It = exp(kD) where D is the gap-width and k is a constant. In these experiments, they show that etching a thin Cu nanowire results in a small gap separation of two electrodes. The size of the nanogap changes in a stepwise fashion, and is restricted to discrete values with changes ∆D that are very close to 0.5 Å . They relate the discrete change on the gap-width to the discrete nature of matter. However, these changes in the nanogap

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99

are considerably smaller than the distances between the low-index lattice planes of the fcc structure of Cu: fcc(111)=2.08 Å (100)=1.81 Å and (110)=1.28 Å . For this reason, Tao and co-workers have postulated that this phenomenom is caused by relaxation processes, proposing that when an atom is added to the surface an atomic reconfiguration takes place towards more stable atomic structure (magic structures) that cause changes in the nanogap much lower than the expected ones (of 2 Å).

Leiva and coworkers have presented the first theoretical analysis of this phenomenon [21]. They have performed energetic calculations and GCMC simulations using the embedded atom method for the interatomic interaction, first they found that the discrete change in the width of the nanogap generates a probability distribution with the most frequently values found in the 0.1-0.4 Å interval. These results were obtained by consideration of the fcc structure of Cu, without the need of assuming atomic rearrangements as proposed by Tao. Leiva and co-workers also show that the values ∆D depends on the relative orientation of the two electrodes. If the orientation of both electrodes has the same low index lattice planes, the ∆D values will correspond to the distance between these lattice planes. On the other hand they show that if a random distribution is chosen, a ∆D distribution results in a maximum value in the range (0.1-0.4 Å). In a second report [22] they show by means GC Monte Carlo calulations that changing the orientation of the cluster with respect to the surface randomly, and take then the probability to obtain a given ∆D they found a maximum close to 0.5 Å . They do not take a tip-like shape to simulate the STM used by Tao and co-workers, instead they have used a cluster with different surface planes in order to reproduce different possibilities in one simulation. But we have to note that until now there are no experimental evidence about how is the geometry and surface planes of a STM tip in atomic detail.

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Figure 6.12: Snapshots taken during dissolution of a nanowire at µ=-3.60 eV. In this study, we have taken a different situation in order to understand the experimental results reported by Tao group. We have taken the nanowire shown in figure 6.8-b as the initial configuration, in which we take into account the STM tip-like geometry and a surface with three layers fcc (111) and the corresponding UPD monolayer. The tip and the surface are Au, like the experimental situation and different from the simulation of Leiva and co-workers. Figure 6.12 shows snapshots of the atomic configuration taken during the dissolution of the nanowire. In frame a) the initial configuration is shown, frame b), c) and d) show different stages of the dissolution process. The dissolution starts at the narrow region of the wire. In frame b) we see a monoatomic wire as the last configuration before break. Immediatly after break three atomic planes disappear, as shown in frame

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c), this can be interpreted by the high tension needed to support this meta-stable configuration. The nano-gap is then formed by dissolving layer-by-layer and also from the border. We have computed the gap-width during evolution of the Monte Carlo simulation, taken the minimal distance between the topmost atoms on the surface and the lower atoms adsorbed on the tip. Figure 6.13 shows the gap-width ∆D as a function of MCS.

Figure 6.13: Gap-width change as a function of MCS, the inset figure shows an enlarged scale. ∆D changes in a stepwise fashion, ∆D jumping distance are frequently of the order of the distance between two fcc (111) planes, indicating that the gap dissolves layer-by-layer, as was predicted by Leiva and co-workers. The inset picture in figure 6.13 shows an enlarged scale, showing the fluctuations of the gap-width around a value close 0.4 Å .

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The simulation results shown here are in agreement with those presented by Leiva and co-workers, indicating that if the orientation of both electrodes has the same low index lattice planes, the ∆D values will correspond to the distance between these lattice planes, in this case Cu(111) = 2.08 Å .

Part IV

Chapter 7 A Molecular Dynamic study of metal deposition 7.1 Introduction Electrochemical deposition of metals and alloys involves the reduction of metal ions from aqueous, organic, or fused-salt electrolytes [43]. The reduction of metal ions M +z in aqueous solution is represented by +z + ze → Mlattice Msolution

(7.1)

This can be accomplished via two different processes: (1) an electrodeposition process in which z electrons (e) are provided by an external power supply and (2) another, electroless (autocatalytic) deposition process in which a reducing agent in the solution is the electron source (there is no external power supply involved). These two processes, electrodeposition and electroless deposition, constitute the electrochemical deposition. The deposition reaction presented by equation 7.1 is a reaction of charged particles at the interface between a solid metal electrode and a liquid solution. We can

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divide the process represented by 7.1 into four types of fundamental subjects: (1) metal-solution interface as the focus of the deposition process, (2) kinetics and mechanism of the deposition process, (3) nucleation and growth process of the metal lattice (Mlattice ), and (4) structure and properties of the deposits. From an atomistic point of view of metal electrodeposition, a metal ion M z+ is transferred from the solution into the ionic metal lattice. A simplified atomistic representation of this process is z+ z+ Msolution → Mlattice

(7.2)

This reaction is accompanied by the transfer of z electrons from the external electron source to the electron gas of the metal M . Atomistic processes that constitute the electrodeposition process (eq 7.2), can be seen by presenting the structure of the initial, M z+ (solution), and the final state, M z+ (lattice). Since metal ions in the aqueous solution are hydrated the structure of the initial state in eq (7.2) is represented by [M (H2 O)x ]z+ . Thus, the final steps of the overall reaction, is incorporation of M z+ adion into the kink site, follow by step decoration. Because of surface inhomogeneity the transition from the initial state [M (H 2 O)x ]z+ (solution) to the final state M z+ (kink) [M (H2 O)x ]z+ (solution) → M z+ (kink)

(7.3)

can proceed via either of two mechanisms: • step-edge site ion-transfer: The step-edge site ion transfer, or direct transfer

mechanism, is illustred in figure (7.1a). It shows that in this mechanism ion transfer from the solution takes place on a kink site of a step edge or on any other site on the step edge. In both cases the result of the ion transfer is a M adion in the metal lattice.

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107

• terrace site ion-transfer: In the terrace site transfer mechanism a metal ion is transferred from the solution to the flat face of the terrace region (Fig. 7.1b). At this position the metal ion is in the adion (adsorbed-like) state having most of its water hydratation. It is a weakly bound to the crystal lattice. From this position it diffuses on the surface, seeking a position of lower energy. The final position is a kink site.

Figure 7.1: Schematic representation of step-edge atom transfer mechanism (a), and atom transfer to the terrace site, surface diffusion and incorporation at kink site (b). Commonly, investigations of metal electrodeposition have been carried out by employing electrochemical, chemical, optical and ex-situ surface analytical methods, which give limited integral information on the structural and mechanistic process of the UPD and OPD processes. With the invention of the local probe techniques (STM and AFM), "in situ" studies together with X-ray methods are providing a new powerfull tool to understand these aspects of metal deposition

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on an "atomic" level. Modelling of metal deposition as a thermodynamic process have been undertaken by several authors in the 1970s ([45, 46, 47]), and recently stochastic Monte Carlo and Kinetic Monte Carlo studies have been taken in order to understand metal deposition using more reallistic many body potential for several metals of interest in electrochemistry ([48, 49]). We have developed a new simulation scheme in order to understand the basis of metal deposition which we call "BD-MD simulation with control of the chemical potential", it combines Brownian Dynamics with Molecular Dynamics using many body potentials to describe the metallic bond. Our method is based on the local control - grand canonical method reported by M. Lupkowski and F. van Swol [50].

7.2 The Model Our model system is divided into two parts or subsystems: the solid surface and the liquid-like phase. We represent the real system of ions, as neutral atoms. The solid surface is modelled as a slab of 6 layers arranged on a fcc (hkl) lattice with approximately 400 atoms per layer with the topmost layer mobile and the two layers below fixed in order to mimic the bulk metal field. The interactions between the various atoms are taken with the Embedded atom method version of Johnson (J-EAM), in which many body effects are taken into account as we have seen in detail in chapter 3.

Figure 7.2 shows a representative picture of the simulation cell. The 3D system has been projected onto a plane perpendicular to the surface. Since we consider the surface parallel to the X − Y plane, we have projected this plot onto the

X − Z plane. Periodic boundary conditions are applied on the X − Y plane, and

7.2 T HE M ODEL

109

Figure 7.2: Schematic representation of the simulation cell. In the lower part the electrode surface is show in grey (spheres) and at the upper part solution particles are represented by dark spheres. The dashed circles around particles close the surface represent the region in which a neighbour list is constructed. Note that a schematic random motion is shown.

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are free in the Z direction. The substrate surface is model by means of Molecular Dynamics, in which we have used the Nordsieck fifth-order predictor-corrector algorithm to integrate Newton’s equation of motion. As shown in chapter 3, this algorithm works in three steps: 1) Predictor: from the positions r(t) and their time derivates r 0 (t) ones predicts the same quantities at time (t + δt) using a Taylor expansion; 2) Force calculation: The force acting in each particle is computed for the predicted positions, and the accelerations a = r 00 (t) = F/m will be different from the predicted acceleration. The difference consitutes the ”error signal”. 3) Corrector: The error signal is used to correct the positions and their derivates [2]. The upper part of 7.2 shows the liquid-like phase, this is simulated by metal atoms (red spheres), governed by a Brownian motion in a constant background potential. The solvent particles as water are omitted, and their effects are represented by a combination of random forces and frictional terms. Brownian dynamics (BD) offers one of the simplest methods for following the trajectories of ions or neutral atoms in a fluid. The algorithm for BD is conceptually simple: the motion of the i−th atom with mass mi is governed by the Langevin equation:

mi

dvi = −mi γi vi + FR (t) dt

(7.4)

The terms on the right-hand side of Eq.7.4 describe the effects of collisions with the surrounding water molecules. The first term corresponds to an average frictional force with a friction coefficient given by mi γi (1/γi is the relaxation time constant of the system). The second term, FR (t), represents the random part of the collisions and rapidly fluctuates around a zero mean. The frictional and random forces in Eq.7.4 are connected through the fluctuation-dissipation theorem [55], which relates the friction coefficient to the autocorrelation function of the random

7.2 T HE M ODEL

111

force, 1 mi γ i = 2kT

Z

+∞

< FR (0)FR (t) > dt

(7.5)

−∞

where k and T are the Boltzmann constant and the temperature, respectively. Ermak’s algorithm [2], is an attempt to treat properly both the systematic dynamic and the stochastic elements of the Langevin equation. A simple algorithm of this type, which reduces to the velocity Verlet algorithm is obtained if, on integrating the velocity equation, the systematic force is assumed to vary linearly with time:

r(t + δt) = r(t) + c1 δtv(t) + c2 δt2 a(t) + δr G

(7.6)

v(t + δt) = c0 v(t) + (c1 − c2 )δta(t) + c2 δta(t + δt) + δv G After the selection of the random components δr G and δv G for a given step, the algorithm is implemented in the usual way. The numerical coefficients in Eq.7.6 are: c0 = e−γδt , c1 = (γδt)−1 (1 − c0 ) and c2 = (γδt)−1 (1 − c1 ). Each atom has a nearest-neighbour list containing information about the position of all atoms that can be found with a defined radius r as we can see in figure 7.2. We have implemented two list, one for MD atoms and another for BD particles. In the case of BD particles, when a i-atom find enough neighbours, that is, when it is close to the surface, then this i-atom is switched to the substrate surface phase and then follows classical Molecular Dynamics (adsorption), starting with his previous velocity in the BD phase. At the same time a new random particle is added into the upper part of the liquid phase, which we call this region "local control-grand canonical (LC-GC)", and is shown in figure 7.2 as the upper region of the dotted line. Of course, we check the minimal distance between the particles

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in order to eliminate overlapping between particles.

Figure 7.3: Schematic representation of potential energy curve. The inverse process happends if the activation energy for dissolution is of the order of kT . In this case the i-atom from the surface is switched to the liquid-like phase, and a particle from the upper part is deleted by random in order to keep the chemical potential constant. The particles in the liquid-like phase do not see each others this is the reason why we use a constant background potential to simulate this region and we make this simple creation-deleting scheme. Figure 7.3 shows a schematic potential energy plot, as we see from right to left the free particles are moving in the Brownian dynamic subsystem with the constant background potential energy denoted by a dashed line. As we mention above, when an i-atom from the liquid-like phase passes the vertical line represented on the plot this atom is switched to the substrate surface phase and therefore "adsorbed". The same happends, if the activation energy is of the order of

7.3 S IMULATION

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RESULTS

kT , an ad-atom can dissolve going to the background potential (solution). As we know from thermodynamics, at equilibrium the Gibbs free energy and therefore the chemical potential for both phases are the same, and can approximate by: 3 V Eb + 3kT = µ0 + kT + kT ln 2 V0

(7.7)

where: µ0 = µ00 +ze∆φ. The left side of equation 7.7 represents the chemical potential of the solid phase, Eb is the cohesive energy of the metal and the second term of the left side represents the entropic effect (k is the Boltzmann constant and T the temperature). The right side of the (eq. 7.7) represent the chemical potential of the solution, the relation of the logarithm is proportional to the pressure for gas phase, or to the activity or concentration for solutions. µ0 is the standard chemical potential, that is related to the difference between the potentials of both phases, that in equilibrium is equal to zero. Then changing ∆φ we can control the deposition or disolution of atoms from the Brownian phase, by means exchange of particles with the solution. When ∆φ > kT deposition is favourable and when ∆φ < kT dissolution is predominant.

7.3 Simulation results Our method permits to explore new scales at the first stages ”atomistic-level” metal deposition and crystal growth, for example: location of the initial atoms, as well fast diffusion processes coupled with formation of surface alloy via-exchange mechanism in (111) or (100) fcc surfaces. One interesting point is to determine the ability of the fluid phase to supply material to the metal surface and the ability of the metal surface to assimilate this material. We can control this process by means of the friction coefficient that is introduced into the Langevin equation. At a given chemical potential µ a particle reaching the electrode surface from the fluid phase

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STUDY OF METAL DEPOSITION

must find a position where it is strongly bound (kink sites or monoatomic steps), otherwise it must return to the solution. The chance of reaching a kink site on the electrode surface depends on the average number of these sites present on the surface, and therefore we can especulate that the growth rate depends on the precise structure of the surface during deposition. Surface diffusion enables a particle to visit several sites on the substrate surface, and will also influence the growth rate. For example a given particle can reach a site very close to a monoatomic step, then this particle can diffuse by means the mechanism showed above (terrace site atom-transfer). We take into account this kind of proccess in order to reproduce as closely as possible the experimental observations.

As a first starting point we have chosen Pt deposition on Au(111) because we have experimental evidence for this system, from electrochemical [51] and UHV [53] experiments. Kolb and co-authors have studied the first stage of Pt deposition onto Au(100) and Au(111). From electrochemical experiments, they found that Pt do not grow layer-by-layer, in contrast to the results reported previously by Uosaki [52]. Insitu STM measurements shows that the electrode surface is free of Pt at a potential of 0.7 V. At a potential of 0.4 V they observed that deposition starts, early nuclei appear preferentially at steps, however soon they appear also on terraces. When an equivalent of about 1 monolayer (ML) was deposited, the cluster had a height of about 0.6-1 nm (3-5 atoms) and a diameter of 10-20 nm. In conclusion, 3D growth was observed in this experiments, followed by Volmer-Weber growth. On the other hand UHV experiments of Pt deposition on Au(111) using STM and Temperature Programmed Deposition (TPD) were performed by Pedersen et.al [53]. They found mainly that Pt deposition starts with the formation of a surface alloy, in which 3% of the Au atoms are replaced by Pt. Subsequently the Pt

7.3 S IMULATION

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115

atoms are deposited on top of the island formed by the Au atoms displaced from the first layer of the substrate. Deposition also in this case starts preferentially at the steps followed by flat terraces. They observed that before completation of the first layer, the second layer starts to grow. The controversial experimental results stimulate to as to study this problem with an atomistic computer simulation point of view. We have performed simulations in perfect flat surfaces, islands and monoatomic steps defects in order to reproduce the experimental situations.

7.3.1 Pt on Au(111) The system consists of an electrode surface that was modeled by six layers of gold atoms arranged in a fcc(111) geometry; the top four gold layers are mobile, and the last two were fixed in order to simulate their bulk properties. Each layer consists of 400 Au atoms and periodic boundary conditions were applied in the X − Y direction. On the upper part of the simulation box (solution) 30 Pt atoms were placed at random positions taken into account no overlapping of the particle

at the initial state.. The interactions between the various atoms were calculated using the embedded atom method, following the instruction and parameters for Au and Pt reported by Johnson (Appendix A). In all cases the temperature was 300 K. Figure 7.4 shows a serie of snapshots taken during deposition. Frame a) shows the inital state with the bare Au(111) surface, frame b) an intermediate state in which we can appreciate a few Pt atoms into the surface, and c) the final state of a flat Au(111) electrode surface after 10 ns, at a chemical potential of -2.0 eV in which deposition from the fluid phase is favorable at all sites by means exchange with Au atoms. Clearly we can observe the tendency of Pt atoms to form 3D crystalline (clusters). The cohesive energies of both metals are (Ec−P t = −5.77eV and

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STUDY OF METAL DEPOSITION

Ec−Au = −3.93eV ) indicating clearly the strong interaction between P t atoms.

The lattice parameter of both metals are (aP t = 3.92 and aAu = 4.08 ), however as ΦM e−S  ΦM e−M e 3D phase formation takes place regardless of the M e − S misfit acording to the Volmer-Weber growth mechanism.

Figure 7.4: Snapshots taken during depostion of Pt on Au(111) at µ=-2.0 eV. Note that the fluid atoms are not included on the plot. As we can note in figure 7.4 a percentage of Pt atoms were exchanged with Au atoms from the electrode surface. This exchange mechanism is a very fast process in which reciprocal segregation of Pt into the Au surface was observed. In order to gain information about this process the evolution of a single Pt adatom was followed by MD monitoring the potential energy of this particle. The result is in agreement with the simulations of Pt deposition, because reciprocal segregation was quickly observed. Figure 7.5 shows the potential energy (Epot) for the Pt ad-atom as a function of the MD steps. The Pt ad-atom was positioned at the beginning at a distance rm bigger than the cutoff distance used in EAM. The (dashed) line shows the potential energy when the ad-atom adsorbed on a monoatomic step. Note for example how the energy decrease when the particle is attracted by the surface. The (full) line represent the potential energy of the adatom close to the flat surface, without influence of any step or kink site. Clearly the plot shows that a exchange between Pt-Au is energetically more stable than direct deposition on the monoatomic step.

7.3 S IMULATION

117

RESULTS

-1 adatom (flat) adatom (step)

EPot [eV]

-2

-3

-4

-5

0

1000

2000

MD Steps

3000

4000

5000

Figure 7.5: Potential energy of the Pt ad-atom as a function of MD steps. Dashed line: for deposition on a monoatomic step, full line: deposition on a flat surface (terrace) We have performed also simulations on electrodes with surface defects. For example Fig.7.6 shows the final state of the surface with a monoatomic step after 10 ns at two different chemical potentials: µ=-2.5 eV (left) and µ=-2.0 eV (right). At -2.5 eV tree dimensional cluster are not observed, however at -2.0 eV 3D crystalline start to grow. Reciprocal segregation of Pt-Au is observed like in the case of a bare surface, and deposition on steps as well flat terraces occurs at the same time, as predict by the potential energy curve. Simulations with islands on the surface as a defect, reveal the same results, exchange of Pt atoms with Au, and depostion of Pt atoms close to the island. Figure 7.7 shows a typical initial and final configuration for this kind of system. The agreement of these results with those reported by Gimenez (et.al) [49] is remarkable. In that paper 2D-lattice Kinetic Monte Carlo reveals surface alloy formation of Pt depostion on Au(100) and Au(111) using EAM potentials. Also

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STUDY OF METAL DEPOSITION

Figure 7.6: Final configuration after deposition of Pt atoms on Au(111) with surface defects (monoatomic step) at two chemical potentials: (left) µ=-2.5 eV with 19 Pt atoms and (right) µ=-2.0 eV with 37 Pt atoms

Figure 7.7: Initial and final configuration after depostion of 29 Pt atoms on Au(111) with surface defects (island) at -2.0 eV

7.3 S IMULATION

RESULTS

119

they showed formation of alloy into islands as we have observed with our model. Finally these simulation results are in agreement with the alloy model reported in the UHV experiments, in which they proposed a mechanism in which the Au atoms expelled from the surface form small island onto Pt.

7.3.2 Ag on Au(111) The system Ag/Au(111) is a typical example for metal (Me) underpotential deposition (UPD) on foreign substrate surface (S) with strong Me-S interaction but negligible Me-S misfit. Ag UPD on Au(111) occurs within a potential range where a surface reconstraction of the substrate can be excluded [56]. At low underpotentials Au(111) − (1x1)Ag structures where observed by in-situ STM [57]. Alloy formation between Ag and Au(111) can be neglected in the UPD range [56]. STM

images also reveal 2D nucleation and growth at monoatomic steps. We have performed 1 a series of simulations with the same computational detail presented in previous section. The same Au(111) substrate surface was used as well the EAM potential to describe the metallic bond. The silver atoms were positioned at random position on the BD phase, like in the case of Pt/Au(111) system. Figure 7.8 shows the final configuration of the Au(111) surface with a gold island on the flat substrate, after Ag deposition during 10 ns of dynamic simulation. At µ = −1.9eV Ag deposition starts only at kink sites as we can appreciate

in Fig.7.8-c). Shifting the potential more positive µ = −1.7eV , silver deposition

start also at kink and monoatomic steps as shows Fig.7.8-b) and finally at a chemical potential µ = −1.3eV deposition of Ag start at steps as well on flat terrace

as shows the final configuration in Fig.7.8-a). A potential energy plot of one Ag (ad-atom) on Au(111) like figure 7.5 reveals energetically favourable deposition 1

In collaboration with Kay Pötting, Abt.Elektrochemie, Universität Ulm. (Some pictures pre-

sented for this system were edited by Kay)

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STUDY OF METAL DEPOSITION

Figure 7.8: Final configuration after depostion of Ag on Au(111) with surface defects (island) at three chemical potentials (µ). a) -1.3 eV, b)-1.7 eV and c) 1.9eV on the following order: kink > steps > terrace. As we mentioned above the introduction of surface defect is a very important parameter to control deposition, for that, simulations on flat surfaces with a monoatomic step were also performed. Figure 7.9 shows the final configuration at three different chemical potentials.

Figure 7.9: Final configuration after depostion of Ag on Au(111) with surface defects (steps) at three chemical potentials (µ). a) -1.5 eV, b)-1.7 eV and c) 1.9eV At µ = −1.9eV silver deposition starts at kink sites also, as shown frame (c),

and at potentials more positive (µ = −1.7eV and µ = −1.5eV ) deposition start preferencially at monoatomic steps as well at the terraces as shows Fig.7.9a-b).

The arrival of particles from the solution is one of the main process, and we

7.4 A

FEW CONCLUSIONS ON METAL DEPOSITION

121

can control this rate, by means of the friction coefficient (γ). Large γ decreases the arrival of atoms, while small values of γ increase the deposition rate. We have used γ = 30, in order to get a smoothly constant deposition rate. Finally, is important to remark that the deposition of a complete monolayer (ML) require approximately, between 7-10 millions MD steps that in a modern computer can take 2 months, therefore simulations with a complete monolayer are not shown, because we are interested on the first stage of metal deposition.

7.4 A few conclusions on metal deposition We have developed a new simulation scheme in order to study metal deposition. The model was tested for two systems: Pt on Au(111) and Ag on Au(111). For both system we found agreement with experimental observations, as well with Kinetic Monte Carlo calculations. In the first case, we have observed a reciprocal aggregation of Pt-Au, followed by 3D crystalline growth, results that are in accordance with the experimental results [53]. In the second case Ag/Au(111), we have observed a 1x1 structure of Ag on Au(111) at low underpotentials, as also deposition preferentially starts at kink and monoatomic steps in agreegment with the experimental results presented by S.Garcia et.al [56]. Finally, this simulation technique could be adapted to any kind of deposition process or crystal growth, new interatomic potentials shall be necessary depending on the system.

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Part V

Chapter 8 General conclusions The main subject of this work involves the application and development of several computational techniques to simulate electrochemical nanostructures and first stages of metal deposition. The simulation techniques employed in the present work were: molecular dynamics, Monte Carlo method and Brownian dynamics, using in all cases the embedded atom method (EAM) for the interactions between the atoms.

In a first part, we study the generation of small metal clusters with the aid of a STM tip, following the Tip Induced Metal Deposition method, reported by Kolb’s group [25]. In these studies, we have analyzed different pairs of metals: Pd/Au(111), Pb/Au(111), Cu/Pt(111) Ag/Pt(111), and arrived of the conclusion that optimal systems for nanostructuring are those where the metals participating have similar cohesive energies and negative heats of alloy formation. We have analyzed also the effect of the crystalline nature of the tip on the nanostructuring process, and we found that the crystalline orientation of the tip plays an important role in this process. When the tip orientation is such that a (111) facet of the tip is facing the surface, the nanostructuring process is accompanied

126

C HAPTER 8. G ENERAL

CONCLUSIONS

by the development of a stronger perturbation of the surface. The force shows a well defined sequence of elastic and plastic changes, which is absent in the case of more open faces, like the (110).

In a second part, we have analyzed the generation of supported metal nanowires by means of molecular dynamics and grand canonical Monte Carlo simulations. Two different methods were used: the mechanical and the electrochemical one. In the first, the mechanical elongation between a STM tip and an electrode surface was studied. We have found a very hard jumps in the z component of the force (Fz ), and a rearrangement of atoms during the elongation process. The minimal cross-sectional area of the nanowire shows a stepwise fashion change during elongation and was related with the jumps on Fz . In the electrochemical case, we have studied the formation and dissolution of metal nanowire using Monte Carlo calculations with control of the chemical potential. On the generation of fcc [111] Cu nanowires, we observed different crystalline planes on the borders on the wire during deposition. In contrast, for [100] Cu nanowires, simulations show a helical growth mode. The formation of molecular gaps between two electrodes was also studied, and we found a step-wise fashion change on the gap-width as was reported on experiments and previous theoretical studies [18, 21, 22].

Finally, we developed a new simulation scheme to be used for metal deposition. The model system is divided into two parts or subsystems: the solid surface, simulated by means of molecular dynamics and the liquid-like phase, simulated by means of brownian dynamics. We represent the real system of ions, as neutral atoms. In the upper part of the simulation cell, we control the chemical potential

127 in order to reproduce the experimental situation of metal deposition. The model was then tested for two systems: Pt on Au(111) and Ag on Au(111). For both system we found agreement with experimental observations, as well with Kinetic Monte Carlo calculations. In the first case, we have observed a reciprocal aggregation of Pt-Au, followed by 3D crystalline growth, results that are in accordance with the experimental results [53]. In the second case Ag/Au(111), we have observe a 1x1 structure of Ag on Au(111) at low underpotentials, as also deposition preferentially starts at kink and monoatomic steps in agreegment with the experimental results presented by S.Garcia et.al [56]. Finally, this simulation technique could be adapted to any kind of deposition process or crystal growth, new interatomic potentials may be necessary depending on the system.

128

C HAPTER 8. G ENERAL

CONCLUSIONS

Chapter 9 Zusammenfassung Der Hauptteil dieser Arbeit beinhaltet die Entwicklung und Anwendung verschiedener Simulationstechniken mit dem Computer für die Untersuchung elektrochemischer Nanostrukturen und der Anfangsstadien der Metallabscheidung. Die folgenden Simulationstechniken werden in dieser Arbeit angewendet: Molekulardynamik, Monte Carlo Methoden, und Brownsche Dynamik. In allen Simulationen wird die "Embedded Atom Method" (EAM) für die Beschreibung der Wechselwirkungen zwischen den Atomen benutzt. Im ersten Teil dieser Arbeit haben wir die Generierung von kleinen Metall Clustern mit der STM-Spitze ("Tip Induced Metal Deposition Method"), Kolb et al. [25]. In diesen Untersuchungen haben wir verschiedene Metallsysteme betrachtet: Pd/Au(111), Pb/Au(111), Cu/Pt(111), Ag/Pt(111). Wir kamen zu dem Ergebnis, dass Systeme mit ähnlichen Bindungsenergien und negativen Enthalpien für die Legierungsbildung optimal für die Nanostrukturierung sind. Wir haben auch die Effekte, welche die Kristallstruktur der STM-Spitze auf den Bildungsprozess von Nanostrukturen ausübt, studiert und fanden, dass die Orientierung der Kristallstruktur auf der STM-Spitze eine wichtige Rolle in diesen

130

C HAPTER 9. Z USAMMENFASSUNG

Prozessen spielt. Wenn die (111) Struktur der STM-SPitze parallel zur Elektrodenoberfläche ist, findet der Nanosrukturierungsprozess unter Entwicklung einer verstärkten Störung der Oberfläche statt. Die Kraft zeigt definierte Folgen elastischer und plastischer Änderungen, welche bei offeneren Oberflächen, wie die (110)Fläche, nicht auftreten. Im zweiten Teil der Arbeit haben wir die Erzeugung gestützter Nanodrähte mit Molekulardynamik und großkanonischer Monte Carlo untersucht. Zwei verschiedene Methoden wurden benutzt: Die mechanische und die elektrochemische Methode. Bei der Untersuchung der mechanische Störung zwischen der STM-Spitze und der Elektrodenoberfläche fanden wir sehr große Sprünge in der z-Komponente der Kraft Fz , und eine Reorganisierung der Atome während der Störung. Die Schnittfläche des Nanodrahtes an der Stelle mit kleinstem Durchmesser zeigt während der Elongation stufenförmige Formänderungen auf, welche mit den Kraftsprüngen Fz zusammenhängen. Im elektrochemische Fall studierten wie die Bildung und Auflösung von metallischen Nanodrähten mit der Monte Carlo Methode im großkanonischen Ensemble. Während der Bildung von Cu(111) Nanodrähten, stellten wir verschiedene kristalline Schichten am Rand des Drahtes während der Abscheidung fest. Im Gegesatz zu den (111)-Drähten zeigten die (100)-Drähte ein schraubenförmiges Wachstum. Die Bildung molekularer Lücken zwischen den beiden Elektroden wurde ebenfalls studiert, und wir fanden eine stufenförmige Änderung der Lückenlänge, wie schon in Experimenten und vorangegangenen theoretischen Arbeiten berichtet wurde.

Schließlich haben wir eine neue Simulationstechnik für die Metallabscheidung entwickelt. Das System ist in zwei Teilsysteme unterteilt: Die Metalloberfläche, welche mit klassischer Molekulardynamik behandelt wird, und die flüssige Phase,

131 in der wir eine Lösung mit Brownscher Bewegung simulieren. Diese Modell wurde für zwei Systeme getestet: Pt/Au(111) und Ag/Au(111). Die Simulationen für beide Systeme waren in guter Übereinstimmung mit den Experimenten und mit kinetischen Monte Carlo Simulationen. Im System Pt/Au(111) haben wir in Übereinstimmung mit den experimentellen Ergebnissen [53] einen Austauschmechanismus zwischen den ersten Goldatomem und den Platinatomen gefunden, gefolgt von 3D Wachstum von Platin (Volmer-Weber). Im zweiten System Ag/Au(111) fanden wir bei niedrigen Unterpotentialen eine 1x1-Struktur von Ag auf Au(111). In Übereinstimmung mit den Experimenten und den Untersuchungen von Garcia et al fonden wir [56], dass die Abscheidung bevorzugt an Halbstufen und an monoatomaren Stufen stattfindet. Diese Arbeit wurde in der Abteilung Elektrochemie der Universität Ulm angefertigt. Marcelo Mariscal, Ulm 2004

132

C HAPTER 9. Z USAMMENFASSUNG

Part VI Appendix

Appendix A

Cu

Ag

Au

Ni

Pd

Pt

Al

Pb

re

2.5561

2.8918

2.8850

2.4887

2.7508

2.7719

2.8861

3.4997

fe

1.5544

1.1062

1.5290

2.0070

1.5954

2.3365

1.3923

0.6478

ρe

22.1501

15.5392

21.3196

27.9847

22.7705

34.1088

20.2265

8.9068

α

7.6699

7.9445

8.0861

8.0296

7.6050

7.0799

6.9424

8.4684

β

4.0906

4.2370

4.3126

4.2824

4.0560

3.7759

3.7026

4.5164

A

0.3275

0.2660

0.2307

0.4396

0.3854

0.4496

0.2515

0.1348

B

0.4687

0.3862

0.3366

0.6327

0.5451

0.5937

0.3133

0.2030

κ

0.4313

0.4253

0.4207

0.4134

0.4255

0.4134

0.3951

0.4258

λ

0.862

0.8507

0.8415

0.8268

0.8511

0.8269

0.7902

0.8517

Fn0

-2.1764

-1.7296

-2.9302

-2.6939

-2.3204

-4.0995

-2.8067

-1.4196

Fn1

-0.1400

-0.2210

-0.5540

-0.0660

-0.4212

-0.7547

-0.2761

-0.2286

Fn2

0.2856

0.5415

1.4894

0.1704

0.9665

1.7665

0.8934

0.6300

Fn3

-1.7508

-0.9670

-0.8868

-2.4574

-0.9326

-1.5782

-1.6372

-0.5609

F0

-2.19

-1.75

-2.98

-2.70

-2.36

-4.17

-2.83

-1.44

F1

0

0

0

0

0

0

0

0

F2

0.7029

0.9839

2.2838

0.2822

1.9662

3.4747

0.9295

0.9210

F3

0.6837

0.5209

0.4941

0.1028

1.3967

2.2883

-0.6823

0.1088

η

0.9211

1.1494

1.2869

0.5098

1.3997

1.3934

0.7792

1.1723

Fe

-2.1916

-1.7512

-2.9813

-2.7004

-2.3626

-4.1743

-2.8294

-1.440

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C HAPTER A.

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Acknowledgments

• I wish to express my gratitude to my supervisor Wolfgang Schmickler, be-

cause his powerfull motivation to work with great ideas and also because his hospitality during my stay in Ulm.

• My great thanks is for my wife, because without her, this thesis would not be possible. At the same time (muchas gracias a mi familia que desde Argentina me enviaron siempre mucha fuerza) • A special thanks also goes to Mario Del Popolo and Ezequiel Leiva (from

University of Cordoba - Argentina) because they were my second guides. Thanks Mario also for some pictures and the gcmc code.

• I would like to thank the Deutsche Forschungsgemeinschaft (DFG) for the

financial support. Thanks also to Prof. M. Naka (University of Osaka, Japan) for financial support during my stay in Japan.

• My thanks also to Kay Pötting by the shared hours in our project of "gcmd", for the nice discussions and the friendship!. Thanks also for the thesis read-

ing and part corrections. • Finally I would like to thank everybody who helped me in my first days in Ulm (Edda, Stefan, Anna, etc.)

Curriculum Vita Personal Data: • Name: Marcelo Mario Mariscal • Birthday: 21.05.1975 • Birthplace: Urdampilleta, Buenos Aires province, Argentina • Marital status: married Education: • (1989-1994): First high-school ENET II Neuquen, Argentina. • (1995-2001): Licentiate in Chemistry, University of Cordoba, Argentina. • (2001-2004): PhD student in the department of Electrochemistry, University of Ulm, Germany.

Positions Held: • 1998-1999: Student teaching assistant, Department of Physical Chemistry, Faculty of Chemistry, University of Cordoba, Argentina.

• 1999-2001: Fellow of the Department of Mining and Geochemistry of Cordoba, Argentina.

• 1999-2001: Fellow of the University of Cordoba, Argentina. • 2000-2001: Teaching assistant, Unit of Math and Physics, Faculty of Chemistry, National University of Cordoba, Argentina

• 2001-2003: Teaching assistant, Department of Electrochemistry, University of Ulm, Germany.

Contribution to Scientific Meetings: 13 Contributions to scientific meetings, nationals and internationals. The most outstanding: • "An embedded atom study of metal deposition onto stepped single crystal

surfaces", M. Mariscal, M. Del Popolo and E. Leiva - "51th International Society of Electrochemistry Meeting" - 2000, Varsovia (Poland).

• "Proton binding at clay surface in aqueous media" M. Avena, M. Mariscal and C. De Pauli - "12th International Clay Conference 2001", Bahia Blanca (Argentina). • "Gereration and Stability of Metal Cluster generated with the Electrochemical S.T.M" M.Mariscal M.Del Popolo, E. Leiva and W. Schmickler, "De-

signing of Interfacial Structures in Advanced Materials and their Joints Meeting" - 2002, Osaka (Japan). • "Lead nanostructures on Au(111) generated with EC-STM, a MD approach", M. Mariscal and W. Schmickler, "9th Fischer Symposium" - 2003, TU Mu-

nich (Germany). International Conferences (Attended): • ”Advances in Electrochemistry - nanotechnology ” Günzburg, Germany (2001). • ’’International conference on designing of Interfacial Structures in Advances Materials and their Joints -DIS’02 ” Osaka, Japan (2002).

• ”1st Spring Meeting of the International Society of Electrochemistry” Alicante, Spain (2003)

• ”9th International Fischer Symposium” Munich, Germany (2003).

List of Publications

1. Generation of palladium cluster on Au(111)- experiments and simulations M. Del Popolo, E. Leiva, H. Kleine, J. Meier, U. Stimming, M. Mariscal, and W. Schmickler, App. Phys. Lett, 81 (2002) 2635. 2. Generation and stability of metal clusters generated with the EC-STM - M. Mariscal, M. Del Popolo, E. Leiva, and W. Schmickler, Proceeding of International Conference on Designing of Interfacial Structures in Advanced Materials and their Joints, High Temperature Society of Japan, (2002) pp. 260-265 - ISBN: 0387-1096. 3. A combinate experimental and theorical study of the generation of palladium clusters on Au(111) with a scanning tunnelling microscope - M. Del Popolo, E. Leiva, H. Kleine, J. Meier, U. Stimming, M. Mariscal, and W. Schmickler, Electrochim. Acta 48 (2003) 1287. 4. The basis for the formation of stable metal clusters on an electrode surface - M. Del Popolo, E. Leiva, M. Mariscal, and W. Schmickler, Nanotechnology 14 (2003) 1009. 5. Proton binding at clay surface in water - M. Avena, M. Mariscal and C. De Pauli, App. Clay Sci. 24 (2003) 3. 6. On the generation of metal clusters with the electrochemical scanning tunneling microscope - M. Del Popolo, E. Leiva, M. Mariscal, and W. Schmickler, Surf. Sci (2004) in press. 7. Effects of tip structure on the generation of metal clusters by an STM tip: a way to controle the orientation of nanocristallites. - M. Mariscal, C.

Narambuena, M.G. Del Pópolo and E.PM.Leiva (2004) (submitted to NanoLetters) 8. On the surface properties of lead structures on Au(111), an atom dynamics approach - M. Mariscal, and W. Schmickler. (2004) (manuscript in preparation) 9. A new simulation scheme to study metal deposition - M. Mariscal, K. Pötting and W. Schmickler. (2004) (manuscript in preparation)