Analysis of diffusion paths of an Al adatom via NEB method#

Advance/NanoLabo, an integrated GUI for nanomaterials, enables you to calculate activation energies of chemical reaction processes and diffusion processes via Nudged Elastic Band (NEB) method¹. You can execute NEB calculation easily just by designating initial- and final-state of the reaction process on Advance/NanoLabo.

We will show analysis of diffusion processes of an Al adatom on Al (001) surface via NEB method below.

Note

The NEB calculation function was implemented newly in Advance/NanoLabo ver1.2. For details, please visit the following site: Integrated GUI for nanomaterials: Advance/NanoLabo

NEB method#

NEB method is a method to find the transition path of a reaction of interest when structures of initial- and final-state are given. The energy curve along the transition path obtained by NEB method can be used to estimate the activation energy of the reaction. For example, you can predict the most probable reaction pathway and compare the calculated activation energy with the experimental one about such as surface reactions and surface diffusions by NEB method.

NEB algorithm is as follows. First, some intermediate images between the initial- and final-state are generated. Next, imaginary springs are introduced between intermediate images. Imaginary spring forces are applied along the reaction coordinate. Structural optimization of each image in which energy gradient includes imaginary spring forces is repeated, and images along the transition path of a reaction are obtained finally.

Conceptual diagram of NEB method

Surface diffusion of an Al adatom on Al (001) surface#

We will consider surface diffusion of Al adatom on Al (001) surface.

Two diffusion paths of an Al adatom can be realized as shown in the figure below. One is hopping, in which an adatom hops to a neighboring adsorption site on the surface, and this diffusion path is the most intuitive. The other is exchange, in which an adatom is exchanged with an atom of the first layer of the lattice.

We will calculate activation energies of each diffusion path via NEB method and predict which diffusion path is more probable.



Diffusion by hopping	Diffusion by exchange

Setting up of NEB calculation#

You can execute NEB calculation just by designating structures of initial- and final-state of the reaction on Advance/NanoLabo. If you designate structures of initial- and final-state, structures are interpolated and some initial-images are generated automatically. The images generated automatically can be recognized briefly by visualization function which displays afterimages.

The figure below shows setup screen of NEB calculation² of diffusion process by hopping. We set number of images as nine.


Designating structures of initial- and final-state	Afterimages of interpolated images

Diffusion by hopping#

We show NEB calculation results of diffusion process by hopping. Activation energy of the process was obtained as 0.46 eV


Energy curve	Forces applied to each image

Diffusion by exchange#

We show NEB calculation results of diffusion process by exchange. Activation energy of the process was obtained as 0.14 eV


Energy curve	Forces applied to each image

Comparison of hopping process and exchange process#

Diffusion path	Activation energy (eV)
Hopping	0.46
Exchange	0.14

The NEB calculation revealed that activation energy of exchange process is lower than that of hopping process by 0.3 eV. This result indicates that diffusion by exchange is more probable than that by hopping.

At first glance, it seems strange that exchange process, which involves exchange of an adatom with an atom of the lattice, is more probable than hopping process, which is the most intuitive³. However, this phenomenon has been observed by Field Ion Microscope (FIM) experimentally.

Such diffusion of adatom by exchange has been observed on surface of other metals like Pt and Ir⁴⁵⁶.