Origin of quasi-constant pre-exponential factors for adatom diffusion on Cu and Ag surfaces Handan Yildirim, Abdelkader Kara, and Talat S. Rahman Department of Physics, University of Central Florida, Orlando, Florida 32816-2385, USA Received 20 June 2007; revised manuscript received 1 September 2007; published 18 October 2007 Many-body interaction potentials from the embedded atom method with two functionals and electronic structure calculations based on density functional theory and the plane-wave pseudopotential method are used to calculate the pre-exponential factors for self-diffusion of adatoms via hopping on Cu100 and Ag100 surfaces with and without steps. The pre-exponential factors are found to be in the range of 10 -3 cm 2 /s for all investigated processes regardless of whether substrate vibrational dynamics are included or omitted. When substrate dynamics are ignored, compensation effects between stiffening and softening of the vibrational frequencies of the diffusing atom are responsible for this quasi-constant pre-exponential. When these dynamics are included, subtle cancellations in the vibrational free energy make the local contribution of the diffusing atom the dominant one. DOI: 10.1103/PhysRevB.76.165421 PACS numbers: 68.35.Md, 68.35.Fx, 66.30.Fq, 63.22.+m I. INTRODUCTION Thermally activated processes often control the end prod- uct in technologically important processes such as thin film growth and heterogeneous catalysis. Detailed and accurate knowledge of relevant energetics and dynamics of such pro- cesses is thus essential if simulation of spatio-temporal evo- lution of materials is to have predictive power. One of the major computational techniques used to study such evolution of materials is kinetic Monte Carlo calculating diffusion co- efficients estimated from harmonic transition state theory. 1,2 These coefficients depend on two main ingredients, namely, the activation energy barrier and the pre-exponential factor or prefactor. Much attention has been given to the calcula- tion of the activation energies, while the prefactor is often assumed to take the “standard value” of 10 -3 cm 2 /s. 3–8 It is also customary to note that uncertainties in the activation energies would generate fluctuations in the diffusion coeffi- cient that are much larger than those generated by deviations in the prefactors from the standard value. Since accurate de- termination of the activation energies for example, using density functional theory is becoming more and more fea- sible, focus has been turning toward a more realistic deter- mination of the prefactors. Such knowledge is certainly im- portant for cases in which accurately determined energy barriers for competing processes lie very close in value to one another. In previous publications, 9–11 a detailed description of a quantum mechanical approach to calculate these prefactors within the harmonic/quasiharmonic approximation has been presented and recently applied to the case of adatoms hop- ping on terraces and steps of Cu100 and Cu110. 9 Indeed, the prefactor was found to be of the order of 10 -3 cm 2 / s with a variation of about less than 1 order of magnitude. We should note here that a full quantum mechanical treatment of the prefactor is not a trivial matter even when the interatomic interaction potential is of a semi-empirical nature. 9–11 In such calculations, force constant matrices evaluated from the par- tial second derivatives of the potential for the whole system diffusing entity plus substrate in the minimum energy and saddle point configurations need to be calculated. Conse- quently, if the system has N atoms, it presents 3N modes at the minimum energy configuration and 3N-1 for the saddle point configuration. With these frequencies, or their densities of states, one calculates the prefactors using the recipe pre- sented in previous publications. 10,11 While this procedure is feasible when the interaction potentials are of empirical or semi-empirical nature, it becomes quickly formidable, with increasing system size when the interaction is described us- ing density functional theory DFT. Understandably, calcu- lations of the prefactors based on DFT have been carried out by totally or partially neglecting the dynamics of the substrate. 12,13 As a matter of fact, for the studied fcc metals, these approximations do not appear to be drastic, as shown by Ratsch and Scheffler, for the case of Ag adatom diffusion on Ag111 for which the prefactor changes only by a factor of 2 when the dynamics of the substrate are partially included. 13 There was thus an informal consensus that for most fcc metals the prefactor for adatom hopping was close to the “standard” value and that the dynamics of the substrate played a minor role in its determination. In a recent publication, Kong and Lewis, 14 however, claim that the role of the substrate dynamics is crucial for the determination of the prefactor for self-diffusion on the same set of metal surfaces as above. Note that while previous DFT calculations have included the substrate dynamics partially, in one previous study 9 and see references therein, calcula- tions based on semi-empirical potentials have incorporated the full vibrational dynamics of the substrate in calculating all contributions to the system vibrational entropy. Note also that in previous publications 10,11 while only local contribu- tion to the system vibrational entropy was emphasized, cal- culations nevertheless included full substrate dynamics. We would like to mention that in a recent study 15 of both adatom and dimer diffusion on the 100 and 110 surfaces of Ag and Cu, using interaction potentials based on the embedded atom method 16 EAM, we also find the prefactor to be “normal.” As we shall see, noticeable cancellations and PHYSICAL REVIEW B 76, 165421 2007 1098-0121/2007/7616/16542110 ©2007 The American Physical Society 165421-1