Low-Temperature Surface Diffusion on Metallic Surfaces - American

Feb 24, 2009 - involved in the Cr adatom surface diffusion at temperatures below 40 K. We ... magnetic circular dichroism (XMCD).11 We have revealed t...
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J. Phys. Chem. C 2009, 113, 4461–4467

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Low-Temperature Surface Diffusion on Metallic Surfaces H. Bulou,* F. Scheurer, and C. Boeglin Institut de Physique et Chimie des Mate´riaux de Strasbourg, UMR 7504 CNRS-UniVersite´ Louis Pasteur, 23 rue du Loess, BP 43, F-67034 Strasbourg Cedex 2, France

P. Ohresser and S. Stanescu Synchrotron SOLEIL, L’orme des Merisiers, St-Aubin BP 48, F-91192 Gif-sur-YVette Cedex, France

E. Gaudry LSG2M, UMR CNRS 7584, Institut National Polytechnique de Lorraine, Ecole des Mines de Nancy, Parc de Saurupt, F-54042 Nancy Cedex, France ReceiVed: June 27, 2008; ReVised Manuscript ReceiVed: NoVember 28, 2008

We present a study of the atomic surface diffusion at low temperature of Cr adatoms deposited on Cu(100), Cu(111), and Au(111) surfaces. Time-dependent X-ray magnetic circular dichroism (XMCD), Monte Carlo simulations, and the Kimball-Shortley iterative method are used to evidence a mechanism of quantum tunneling involved in the Cr adatom surface diffusion at temperatures below 40 K. We show that the XMCD line shape is correlated to the atomic configuration of Cr deposited on a metallic surface. A diffusion rate ranging from 0.03 Hz for Cu(100) to 30 Hz for Au(111) at 10 K is reported from the time evolution of the XMCD line shape. A quantum equation of motion for the Cr adatom is derived and solved by using a Kimball-Shortley iterative method. Good agreement between the calculated Bohr periods and the experimental diffusion rates is reported. I. Introduction Quantum tunneling is one of the most intriguing properties of matter and is fundamental in manifold phenomena in physics and chemistry. Among the most outstanding advances allowed by quantum tunneling, one finds atomic imaging1 that led to atomic manipulation and the discovery of quantum mirages,2 cascading surface molecules used to operate elementary nanoscale computers,3 and more recently, quantum fluctuations precluding the formation of reconstructed surface phases.4 Classically, atoms are trapped unless they gain enough energy to overcome the surrounding energy barrier, while quantum mechanics allows a particle to tunnel through the barrier, provided that its de Broglie wavelength5 is large enough. Due to the decrease of the de Broglie wavelength with particle mass and temperature, surface quantum tunneling has so far been observed mainly for light particles. For instance, hydrogen and its isotopes deposited on metal surfaces exhibit a clear transition from thermal overbarrier hopping to quantum tunneling diffusion.6-10 In a previous paper, we have reported an experimental observation of the quantum tunneling diffusion in the case of much heavier surface adatoms by using X-ray magnetic circular dichroism (XMCD).11 We have revealed that, at temperatures below 40 K, the motion of chromium adatoms deposited on Au(111) proceeds by quantum tunneling diffusion. In the following, we will show that the previous results concerning the quantum tunneling diffusion of Cr on Au(111) can be extended to other metallic substrates such as Cu(111) and Cu(100) as well. By employing a variational approach, a quantum equation of adatom motion is derived and solved by using a Kimball-Shortley iterative method.12 By comparing the * Corresponding author. E-mail address: [email protected].

calculations with experimental data, it is evidenced that in the classical regime the relevant parameter is the activation energy, whereas it is the site energy of the adatom in the quantum tunneling regime. We will first describe the experimental setup, and then we present a detailed analysis of the XMCD line shape. Finally, we describe the mechanism of quantum tunneling ruling the atomic diffusion on different metallic surfaces and the specific role of the site energy of the diffusing adatom. II. Experiment The experiment was carried out at beamline ID08 at the European Synchrotron Radiation Facility in Grenoble.13 The metal substrates were prepared by cycles of Ar+ at 300 K and annealing sequences between 800 and 900 K. The growth characterizations were performed in situ by using X-ray absorption spectra (XAS) in an ultrahigh-vacuum chamber (5 × 10-11 mbar) containing a superconducting magnet and a Cr evaporation cell. Cr was evaporated onto clean metallic surfaces between 10 and 140 K. The Cr electron bombardment cell is equipped with a flux monitor measuring the weak ion current of the atom beam. The amount of Cr was estimated from the Cr L2,3 edge jump of the XAS in the 0.1-0.5 ML coverage range,14 allowing a calibration of the flux monitor. During evaporation, there is a sizable drain current on the sample which is proportional to the amount deposited, allowing a cross-check with the fluxmeter indications. The time integration of the drain current gives the amount deposited in a reliable way even in the very low coverage range. The error on the thickness is estimated to be much better than 20%. XAS are obtained by detecting the total electron yield with the light beam aligned along the surface normal. The XMCD signal is measured by recording the XAS at the Cr L2,3 edges for left and right circularly polarized X-rays

10.1021/jp805674n CCC: $40.75  2009 American Chemical Society Published on Web 02/24/2009

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Figure 1. Chromium XMCD spectrum (blue line) of a a collection of isolated Cr atoms on Au(111) (∼0.05 Cr atom/nm2). The red line (open circle symbols) is a multiplet-based calculation for a single Cr atom in a d5 electronic configuration with no crystal field.

Bulou et al.

Figure 2. Cr atomic spectrum (blue line) and high coordinated Cr spectra for Cr deposited on Au(111) (open triangle symbols), Cu(111) (open square symbols), and Cu(100) (open circle symbols) obtained by depositing a density of ∼7 atoms/nm2.

with a 6 T applied magnetic field, parallel and antiparallel to the incoming light. A set of two XAS spectra are measured in about 5 min. During the whole deposition-measure procedure the sample is mounted on a cryogenic sample holder with a cold reservoir at 4 K located in the gap center of the superconducting magnet. III. Results A. Cr Coordination and XMCD Line Shape. The chromium jump rate at low temperatures has been measured using XMCD, a rather unusual method in this field. XMCD measures the difference between two XAS recorded with opposite helicity of the light. It can detect unitary changes in surface adatom coordination15,16 making it ideal to study surface diffusion under specific conditions. The reason for this is that XMCD involves selection rule governed transitions from core levels to unoccupied states just above the Fermi level. These unoccupied states are highly sensitive to the local atomic coordination.17 Whereas single 3d adatoms exhibit sharp multiplet peaks in the L2,3 XMCD spectrum,15 coordinated adatoms exhibit much broader features arising from the band structure. An XMCD spectrum characteristic of isolated Cr adatoms is displayed in Figure 1. It corresponds to a surface density of ∼0.05 Cr atom/nm2 (0.004 atomic layers) deposited on Au(111) at 10 K, a temperature for which one usually assumes that the atoms stick where they land. Random deposition simulations show that in this case 99% of the adatoms are isolated right after the deposition. This low coordinated spectrum is in excellent agreement with atomic multiplet calculations18 for an isolated Cr atom with a d5 electronic ground-state configuration (Figure 1). Absolutely similar Cr spectra are obtained for a deposition in the same conditions on Cu(111) and Cu(100). This low coordinated spectrum will be therefore referenced as Cr atomic spectrum hereafter. It is interesting to note that for single Cr atoms on a noble metal the shape of the Cr L2,3 edges actually does not depend on the substrate, on the chemical nature, or on the surface symmetry. This is corroborated by the multiplet calculation which shows that the best agreement is obtained for a d5 configuration without a crystal field. On the other hand, when the Cr adatoms are not isolated on the surface but form small Cr clusters (we will refer to this situation as high coordinated atoms), the XMCD spectra show some significant changes with respect to the Cr atomic spectrum. Figure 2 shows the previous Cr atomic spectrum together with spectra on various surfaces with a larger amount of deposited Cr (∼7 atoms/nm2). Although there are some differences

Figure 3. Chromium XMCD spectra of a high coordinated Cr atom (∼7 atoms/nm2) deposited on (a) Cu(111) (red line) and Cu(100) (open square symbols) and (b) Cu(111) and Au(111) (open circle symbols).

between the spectra obtained for the three high coordinated situations on different substrates, they appear rather small when compared to the atomic spectrum. A high coordinated XMCD spectrum and an atomic XMCD spectra can be clearly distinguished by the following features as observed in Figure 2: (i) There is a global energy shift of about 2 eV between the Cr atomic spectrum and the high coordinated Cr spectra; (ii) The Cr atomic spectrum is characterized by a very narrow downward structure at the L3 edge followed by small upward structures shortly after the edge, whereas for the high coordinated Cr spectra there is a much broader downward structure at the L3 edge followed by a very broad and intense upward structure after the edge; (iii) The L2 edge of the Cr atomic spectrum is composed of a double upward structure, whereas for the high coordinated Cr spectra it is composed of a downward structure followed by a large upward structure. A closer look on the high coordinated spectra for the various surfaces (Figure 3) shows indeed differences. For instance, a shoulder is observed at the L3 edge for Cr deposited on a Cu(100) substrate, whereas it is absent in the case of a deposit on (111) substrates (Figure 3a). Intensity variations are also observed for the different features when comparing a deposit of high coordinated Cr adatoms on Cu(111) and Au(111) (Figure 3b). Surprisingly small clusters containing only several atoms already present the features of high coordinated atoms in the XMCD spectrum.

Surface Diffusion on Metallic Surfaces

Figure 4. XMCD spectra for a deposit of 0.006 monolayer of Cr on Cu(111) at 10 K recorded 15, 120, and 300 min after the end of the Cr deposition (open square symbols). A linear combination (red line) of the Cr atomic spectrum and the high coordinated Cr spectrum is superimposed for each situation.

B. XMCD Spectrum Decomposition. The differences between an atomic spectrum and a high coordinated one provide a reliable way to characterize the coordination state of deposited chromium atoms whatever the chemical and the structural nature of the substrate. Within a good approximation, the XMCD signal of a given Cr deposit can be decomposed in a high coordinated spectrum arising from clusters and an atomic spectrum arising from the isolated Cr atoms. Indeed, the Cr atomic spectrum and the high coordinated Cr spectra are so different that a Cr spectrum for any situation can be described as a linear combination of these two reference spectra. Examples of such a linear combination are displayed in Figure 4 in the case of a Cr deposit of 0.006 atomic layer on Cu(111) at 10 K. After the end of the deposition, provided that the deposited amount is small enough to have mainly isolated Cr atoms, an evolution as a function of time of the XMCD spectrum is observed. Figure 4 shows that every spectrum is well described by a linear combination of the Cr atomic spectrum and the high coordinated Cr spectrum. Such an approach is applicable for any substrate, provided a suitable high coordinated Cr spectrum is recorded on the corresponding substrate. Having established a reliable procedure to characterize the Cr coordination state, we can focus on the evolution of the Cr XMCD spectra during the time in Figure 4. In the following, we consider only small Cr densities for which the Cr atoms are initially isolated on the surface. A clear evolution from a pure Cr atomic spectrum to a pure high coordinated Cr spectrum is observed as a function of time. Figure 5 displays the time evolution of both atomic and high coordinated contributions, as deduced from a linear combination fit as explained above. A decrease of the atomic contribution and an increase of the high coordinated contribution are observed. After 260 min, the atomic contribution has totally vanished, and the high coordinated contribution has reached a plateau. IV. Discussion A. Atomic Diffusion and XMCD Time Evolution. The time evolution of the XMCD Cr spectra from an atomic feature to a high coordinated one arises from a change of the atomic surrounding of the Cr adatoms. It can stem either from a movement of the Cr adatoms or from a matching of the Cr

J. Phys. Chem. C, Vol. 113, No. 11, 2009 4463

Figure 5. Time evolution of atomic (open circle symbols) and high coordinated (open triangle symbols) contributions of the Cr XMCD spectra in the case of a Cr deposit of 0.006 monolayer at 10 K on Cu(111).

Figure 6. High coordinated Cr/Cu(100) spectra before (black line) and after CO exposure (dashed green line). The red open square symbol curve corresponds to the isolated Cr adatom spectrum, while the blue open circle symbol line gives the XMCD spectrum for a Cr in ruby.

adatoms with other chemical species such as, for example, oxygen or carbon (contamination). Four clues tend to ascribe the time evolution of the XMCD Cr spectra to a diffusion mechanism of the Cr adatoms rather than to a contamination effect. First, the time-evolved XMCD spectra for single adatoms is identical to the high coordinated Cr spectrum, while the exposure of a Cr deposit to several langmuirs of CO immediately at the end of the deposit leads to spectra with pronounced multiplet structures, different from the time-evolved spectra (Figure 6). Second, no evolution is observed at low temperature for situations presenting only coordinated Cr atoms in small clusters. These clusters may only contain several atoms and are obtained, e.g., at higher temperature deposition. Third, increasing the substrate temperature from 10 to 140 K significantly speeds up the time evolution of the XMCD spectra. Fourth, changing the crystallographic structure of the substrate from a (111) surface to a (100) one reveals a lower time evolution of the XMCD Cr spectra, inconsistent with a contamination effect but in agreement with an atomic diffusion process as shown in the following. The ratio of isolated Cr adatoms on both (111) and (100) substrates calculated by a Kinetic Monte Carlo (KMC) procedure as a function of the number of Monte Carlo steps is diplayed in Figure 7. A unit cell of 51 200 adsorption sites has been considered. It corresponds to a unit cell of 46 nm × 40 nm for Au(111), 41 nm × 35 nm for Cu(111), and 58 nm × 58 nm for Cu(100). Due to the small Cr coverages investigated and the low temperature, only single-jump events of isolated Cr adatoms

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Bulou et al. TABLE 1: Duration of Monte Carlo Time Steps and Experimental Jump Rates, Γexp, for a Deposit of 0.6% Cr Monolayer substrate

T (K)

time steps (s)

Γexp (10-2 Hz)

Au(111) Cu(111)

10 10 20 10 20 40 100 140

10.5 3.4 2.0 37.1 78.8 40.5 6.6 2.1

9.5 29.6 50.0 2.7 1.3 2.5 15.1 47.6

Cu(100)

Figure 7. Evolution of the isolated Cr adatom ratio with respect to the number of Monte Carlo steps for both the (111) substrate (straight line) and the (100) one (dashed line).

are significant and have been taken into account. Double jumps and cluster evaporation become important only at elevated temperatures.20 The results presented in this paper arise from an average of a dozen simulations of 105 steps. Figure 7 shows clearly a faster evolution of the ratio of isolated Cr adatoms when deposited on a (111) surface compared with a (100) surface. This is due to a larger number of possible diffusion directions on a (100) surface (four instead of three on a (111) surface) making the probability for an adatom to meet another one smaller on a (100) surface than on a (111) surface. B. Characterization of the Atomic Diffusion. Having established the Cr adatom motion as the mechanism at the origin of the time evolution of the XMCD Cr spectra, we focus now on a quantitative characterization of the Cr diffusion. The classical jump diffusion is the most common mechanism at the origin of adatom motion onto surfaces. In this case, the adatom makes small-amplitude oscillations around the minima localized at the adsorption sites. Occasionally, it gains enough energy from the substrate to make a successful jump, after which it thermalizes again in another adsorption site. This is a thermally activated process, and the temperature dependence of the classical jump rate can be written in the so-called Arrhenius form19,20

( )

Ea Γ ) Γ0 exp kBT

(1)

where T is the temperature; Ea is the activation energy; and kB is the Boltzman constant. Γ0 is a prefactor which contains dynamical quantities; typically Γ0 = 1012-1013 Hz for most surfaces.21,22 The classical jump rate depends exponentially on the temperature; typical activation energies for 3d adatom diffusion on surfaces of transition metal range from 50 to 220 meV for (111) surfaces23-26 and from 280 to 860 meV for (100) surfaces.22,23,27-33 The experimental jump rate is extracted from the XMCD data by adjusting the KMC simulations to the time evolution of the ratio of isolated Cr adatoms by means of a time step factor (Table 1). Figure 8 displays an example of such a fit for 0.006 atomic layer of Cr on Cu(100) at 10 K. The experimental jump rate is in the order of 10-2 Hz whatever the substrate and the temperature investigated in this work. At 10 K, the classical jump rate is ∼10-13 Hz for a (111) surface with an activation energy of 50 meV and ∼10-130 Hz for a (100) surface with an activation energy of 280 meV. These values are much lower than the experimental jump rate and rule out the classical jump rate as a possible mechanism for the Cr motion at 10 K.

An alternative mechanism for diffusion at low temperature is the quantum tunneling diffusion. Indeed, in the considered temperature range, the de Broglie wavelength of a Cr adatom λB ) 2πp/√3mkBT becomes comparable to the site separation distances (p is the reduced Planck constant and m the mass of the Cr atom). Thus, the description of the Cr adatom as a particle located in an adsorption site is no longer satisfactory, the adatom being delocalized on several adsorption sites. In this case, the time evolution of the system needs to resort to quantum methods. The model we used is the following. Let us consider an assembly of atoms, the substrate plus a Cr adatom. Let H and Ψ rNatom) be, respectively, the Hamiltonian and the wave (r b1 · · ·b function of the system. Ψ depends on the coordinates of all atoms of the system, and H is written as Natom

H)

∑

Natom Natom

Ti +

i)1

∑ ∑ Vij

1 2 i)1

(2)

j*i

with

Ti ) -

p2 b 2 ∇ 2mi i

(3)

The first term of eq 2 is a sum over single-atom kinetic operators (mi is the mass of atom i), and the second term is the interatomic interaction between the atoms. As a first approximation, we write Ψ as Natom

Ψ(b r1 · · · b r Natom) )

ψ(b r i) ∏ i)1

(4)

This approximation is fully justified in the framework of this paper because, due to the small de Broglie wavelength, the atomic wave function is mainly localized around the equilibrium position of the atoms in the lattice, except for the Cr adatom.

Figure 8. Time evolution of the ratio of isolated Cr adatoms for a deposit of 0.006 monolayer on Cu(111) at 10 K (circle) and corresponding Kinetic Monte Carlo simulation (solid line).

Surface Diffusion on Metallic Surfaces

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We start by inserting the function (4) into the Schro¨dinger equation HΨ ) EΨ and calculate the expectation value of the energy E ) . Since H reduces to a sum of singleatom and two-atom operators, the matrix elements become products of integrals or and (k * i, j). Assuming that the ψi’s are normalized, the latter become unity, and we are left with Natom

E ) < Ψ|H|Ψ > )

∑

< ψi|Ti|ψi > +

i)1

Natom Natom

∑∑

1 2 i)1

< ψiψj|Vij|ψiψj > (5)

j*i

This is the expectation value of the energy for arbitrarily given ψi’s. In line with the variation principle, those ψi’s which minimize E represent the best set of functions for the ground state.34 By introducing the Lagrange multipliers εi, we add the normalization conditions to eq 5 and carry out the variation

[

δ E-

]

Natom

∑ εi( -1) i)1

)0

(6)

Equation 6 must be valid regardless of the variation δψ*i . It follows that the ψi’s are defined by equation

[

-

]

p2 b 2 ∇ + ViS(b) r ψi(b) r ) εiψi(b) r 2mi

ViS(b) r )

1 2

(7)

Natom

∑ ∫ |ψj(rf′)|2Vij(b,r fr′)drf′

(8)

j*i

Equation 8 suggests that |ψj(r b′)|2dr b′ represents the probability to find the atom j in a volume dr b′ around b′. r In this framework, the Lagrange multiplier εi is the vibrational energy of atom i subject to a potential resulting from its interaction with the other b, t) is the vibrational wave function atoms of the system, and ψi(r of the atom i. b) of eq 7 represents the interaction energy of The term VSi (r the atom i with the other atoms of the system. This can be regarded as a site energy term. Here one must emphasize the b), involved fundamental difference between the site energy VSi (r in the quantum diffusion process, and the activation energy Ea at the origin of the classical diffusion described by the Arrhenius equation (eq 1). The activation energy has a global nature since it takes into account the overall atoms of the system, substrate plus adatom. The site energy represents the single adatom energy in interaction with the atoms of the substrate and is therefore of local nature. It means that the quantum diffusion is mainly determined by the local interaction of the adatom with its closest neighborhood, whereas the classical diffusion is governed by a nonlocal feature, the activation energy. Three examples of the site energy VSi (r) of an adatom moving on Cu(100), Au(111), and Cu(111) are displayed in Figure 9 in the case of homoepitaxy. The site energies are calculated by using a Nudged Elastic Band method26,35 that proved efficient to find the minimum energy path between given adsorption sites on surfaces. The interatomic interactions are based on effective potentials obtained in the framework of the second moment approximation of the tight-binding theory. This framework has an extensive record of reliability for the modeling of transition metal surface and cluster properties, as demonstrated in previous work.28,36-38 The site energies displayed in Figure 9 have been calculated in the case of homoepitaxial systems. Three param-

Figure 9. Site energy for Cu(100), Au(111), and Cu(111) calculated by using a nudged elastic band method. The site energies have been scaled to their maximum, as a function of a scaled intersite equilibrium distance to compare the shapes.

eters allow us to fully characterize the site energy, as observed from Figure 9: the energy V0 at the saddle point, the distance a between two adjacent adsorption sites, the shape of the site energy. The two latter parameters depend mainly on the substrate, while the energy V0 at the saddle point depends on both the chemical nature of the adsorbant and the substrate. V0 ranges from 20 to 100 meV depending on the chemical nature of the adatom and the substrate. To compare the different shapes as a function of the substrate, the site energies have been scaled at the energy maximum and represented as a function of intersite equilibrium distances. One can notice the shape difference between fcc(111) and fcc(100) surfaces, the first one having either fcc or hcp available adsorption sites. Equation 8 allows us to rank the atoms in three classes. The first class encompasses the bulk atoms. Due to their surrounding neighbors, the physical situation of these atoms is equivalent to the situation of a particle in an infinite quantum well for which the wave function remains localized at the initial position. There is no time evolution of the wave function, hence no atomic diffusion for these atoms. The second class is formed by the substrate surface atoms. Due to the missing top neighbors, the surface atoms are immersed in a well with a finite height. However, the barrier is high and the well can be considered infinite. This is further emphasized in the case of the Au(111) substrate since the mass of gold atoms is about four times larger than the Cr one. It means that the de Broglie wavelength is about half of the Cr one, and then, the substrate atoms can be considered as classical particules. The adatoms deposited on the surface form the third class. These atoms have neither inplane nor top neighbors. The adatoms are then immersed in wells with a finite height like the surface atoms but with a much smaller barrier. At the temperatures investigated in this paper, the vibrational energy of an adatom is much smaller than the saddle point energy V0. Each basin of the potential experienced by a Cr adatom is therefore characterized by discrete vibrational energy levels. The degeneracy of the system is equal to the number of basins since the adsorption sites are nearly identical on the surface. Due to the delocalization of the Cr wave function, the discrete vibrational levels are coupled. At the temperatures investigated in this work, the range of the de Broglie wavelength is one interatomic distance, meaning that only two basins are involved in the coupling. The main effect of the coupling between the two basins is to lift the degeneracy which leads to a wave function oscillation between the two basins, leading to tunnel jumps between two adjacent basins. The time evolution of the atom is governed by the equation of motion

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ip

(

)

∂ψi(b, r t) p2 b 2 ∇ + ViS(b) r ψi(b, r t) ) ∂t 2mi

Bulou et al.

(9)

and the period of the oscillation is given by the Bohr period

τBohr )

h ∆ε

(10)

h is the Planck constant and ∆ε the energy difference due to the coupling. Figure 10 gives the evolution of the Bohr period τBohr as a function of the amplitude of the saddle point V0 of the site energy and taking the intersite distances of Cu(100), Cu(111), and Au(111). The Bohr period τBohr is calculated by using the Kimball and Shortley method,12 a finite-difference method to b) solve the Schro¨dinger equation (eq 7). The wave function ψi(r is obtained through an iterative relaxation scheme. Figure 10 evidences the major role of the shape of the site energy in the diffusion process: Assuming artificially an identical saddle point energy V0, although the intersite distance is smaller on Cu(111) than on Au(111), the diffusion is faster on Au(111). This effect is emphasized when the saddle point energy V0 increases. The shape of the site energy depends on the ability of the atoms belonging to the surface to “follow” the adatom when moving from an adsorption site to another one. A reliable criterion for this is the surface mismatch ms,37 which characterizes the difference between the interatomic equilibrium distance in the bulk with respect to the one at the surface. The surface mismatch is about -3% for Au(111) and -2% for Cu(111). It means that the ability for the surface atoms to contract the interatomic distances, in order to “follow” the adatom motion, is much larger at gold surfaces than at copper ones. It results in a much flatter bottom of the well in the case of gold than for copper, inducing a smaller barrier width for Au(111) than for Cu(111). C. Influence of the Temperature: Crossover from Quantum Regime to Classical. Let us turn now our attention to the crossover from a quantum to a classical regime. The exponential form of the classical jump rate leads to a very strong temperature dependence: a very high jump rate at high temperature and a vanishing one at low temperature. In subsection 4.2, a typical classical jump rate for a (111) surface at 10 K has been estimated to ∼10-13 Hz which corresponds to a jump every 300.000 years. For the same situation, at 300 K, the classical jump rate is ∼1012 Hz, that is one jump every picosecond. In the quantum regime, for heavy atoms such as 3d metals, the temperature dependence of the jump rate lies mainly on the

Figure 10. Bohr period for Cu(100) (open triangle symbols), Cu(111) (open circle symbols), and Au(111) (open square symbols) as a function of the site energy. The dashed lines give the value of the respective experimental Bohr times.

Figure 11. Experimental jump rate as a function of the temperature in the case of the Cu(100) substrate.

temperature dependence of the de Broglie wavelength. The higher the temperature, the weaker the delocalization of the adatom and the smaller the degeneracy lifting. A large degeneracy lifting at low temperature allows a high jump rate. Due to the vanishing thermal diffusion, the quantum tunneling regime is dominant at low temperatures and decreases slowly with a stronger localization upon increasing temperature. At a given temperature, the crossover temperature, the classical thermal diffusion rate becomes equivalent to the tunnel one. Above the crossover temperature, the classical regime becomes dominant by orders of magnitudes due to the exponential Arrhenius law. This behavior is clearly demonstrated in Figure 11 which displays the experimental jump rate as a function of the temperature in the case of Cr on Cu(100). A decrease of the jump rate is first observed down to 40 K followed by a very fast acceleration, corresponding to the exponential behavior of the Arrhenius law. The crossover temperature from a classical to quantum regime lies therefore between 20 and 40 K. This value is smaller than TH/Cu(001) ) 60 K reported by Lauhon and Ho in the case of H on Cu(001),6 but not as much as one would expect from the mass ratio between hydrogen and chromium. However, the transition temperature depends on the relative jump rates in the quantum regime and the thermal one: the lower the quantum tunneling rate is compared with the thermally activated diffusion one, the smaller the crossover temperature. For hydrogen, a quantum tunneling rate 60 times smaller than for Cr is reported, whereas the thermal diffusion is about 30 times faster at 80 K,6,39 which explains the small difference between the Cr and H crossover temperatures. The low quantum tunneling rate of H compared with the Cr one deserves to be commented on. Actually, due to the low mass of hydrogen, a large quantum tunneling rate is expected. However, the time evolution of an adatom wave function is ruled by two terms, a kinetic term and a site energy term (eq 9). The kinetic term, at the origin of the wave function propagation, is inversely proportional to the mass of the adatom. The smaller the mass, the higher the quantum tunneling rate is. The site energy term slows down the quantum tunneling rate. Thus, the quantum tunneling rate results from a subtle balance between the kinetic term and the site energy term. For a tunnel diffusion to occur, the kinetic term must be at least slightly larger than the site energy one. In the case of hydrogen (H) and deuterium (D), a quantum diffusion is observed for H but not for D above 40 K. Thus, the site energy termswhich is the same since it uniquely results from the electronic structuresis comprised between the kinetic term of D and the H one, which differ by a factor two (the mass ratio). Since the kinetic terms

Surface Diffusion on Metallic Surfaces of H and D are large because of a low mass, the site energy term is also large. The validity of this argument is supported by a calculation by Herrero40 showing that the thermal diffusion of D is still dominant at 40 K. For some heavier adatoms, such as Cr, although the kinetic term is weak, the site energy term can become much smaller than the one for H and D. The tunnel diffusion can thus occur, provided the temperature is low enough, first to block thermal diffusion and, second, to meet the de Broglie wavelength condition. An estimation of the activation energy, Ea, can be obtained from the jump rate evolution with the temperature above 40 K displayed in Figure 11. By using the Arrhenius equation (eq 1) with a prefactor Γ0 ) 5.0 × 1012 Hz, an activation energy Ea = 360 ( 20 meV is found for the classical diffusion of a Cr adatom on Cu(100). This value is in rather good agreement with the activation barrier for 3d adatom diffusion on Cu(100) surfaces, comprised between 350 and 650 meV.22,33,41,42 For Cr/Au(111), we determine an activation energy Ea ) 100 ( 15 meV, in excellent agreement with the theoretical value of 98 meV obtained via the nudged elastic band method in the framework of the modified embedded atom method.11 Concerning Cr/Cu(111), the situation is more difficult since the thermally activated regime already starts at a very low temperature. It is therefore only possible to give a rough estimation of the activation energy, which is in the order of 30 to 50 meV, which is nevertheless in good agreement with DFT calculations of Bogicevic et al.43 On Cu(111), it is therefore difficult to observe a pure quantum tunneling regime since the thermal regime is already activated at very low temperature giving a thermal diffusion coefficient comparable to the quantum tunneling one. The observation of Cu dimer diffusion on Cu(111) has however been reported and attributed to thermally assisted quantum tunneling.44 V. Conclusion In summary, the sensitivity of XMCD to the surface atomic coordination was used to study low-temperature diffusion of Cr adatoms on various noble metal surfaces. The analysis of the time evolution of the XMCD spectra via Monte Carlo simulations allowed the determination of the jump rates, ranging from 0.03 Hz for Cu(100) to 30 Hz for Au(111) at 10 K. A quantum equation of motion for the Cr adatom using a Kimbal-Shortley iterative method allowed us to calculate the Bohr period for the tunneling diffusion which has been found in good agreement with the experimental jump rates. It is shown that the important parameter governing the tunnel jump rate is not the activation energy barrier but the site energy which is mainly characterized by the intersite distance and the saddle point energy. References and Notes (1) Binnig, G.; et al. Phys. ReV. Lett. 1982, 49, 57. (2) Manoharan, H. C.; Lutz, C. P.; Eigler, D. M. Nature 2000, 403, 512.

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