Point Defects and Hopping

Jin Zhaohui（金朝晖）

http://cms.sjtu.edu.cn/gs

Nov 6, 2019

Hands-on #1

A point defect in a crystal is

• the occupancy of a lattice sites by impurity atoms/ions or a voids (i.e. vacancy);

• extra atoms/ions not in regular lattice positions.

Unlike dislocations, both vacancies and interstitials can be produced by thermal processes and

can exist in equilibrium concentrations governed by their energies of formation.

Their energetics, studied both experimentally and through various forms of simulation, can

involve many complexities due primarily to local atomic relaxations immediately around them in

the case of vacancies and relatively long-range elastic misfit fields in the case of interstitials.

Point defect

G

Δ𝐺#

x

A vacancy occurs at or around a single lattice point and involves a missing atom.

An important property of vacancy is its vacancy formation energy, EVFIt is the energy required to break the bonds between an atom and its nearest neighbor atoms

and removing that atom to where it has no interaction with the remaining system.

represent the final energy after the atom is removed from the cell,

the initial energy before the atom was removed,

the total number of atoms in the cell with the vacancy,

the total number of atoms in the cell without the vacancy.

Part I. Vacancy Formation Energy (T = 0 K)

𝐸%& = 𝐸()*+, −𝑁()*+,𝑁)*)/)+,

𝐸)*)/)+,

𝐸()*+,𝐸)*)/)+,𝑁()*+,𝑁)*)/)+,

1. Generate a 864-atom FCC supercell with 6 x 6 x 6 unit-cell dimension

2. Identify the number of atoms Ninitial in the supercell and calculate the initial energy Einitial

3. Remove one atom from the supercell to create a vacancy

4. Relax the supercell using energy minimization (conjugate gradient or fire)

5. Identify the number of atoms Nfinal in the supercell and calculate the final energy Efinal of the relaxed system containing a vacancy

6. Calculate the vacancy formation energy, EVF (units in eV)

MD: Vacancy Formation Energy (T = 0 K)

dump.0 perfect crystal (864 atoms)

dump.1 vacancy (863 atoms)

dump.2 vacancy after minimization (863 atoms)

𝐸%& = 𝐸()*+, −𝑁()*+,𝑁)*)/)+,

𝐸)*)/)+,

define a group of one atom (id = 476)

create a 6x6x6 fcc crystal, units in bins (4 atoms/bin)

Energy minimization

delete the group to generate a vacancy

model material: Cu, EAM

periodic boundary conditions input file for vacancy

𝐸%&0 = 𝐸()*+, −𝑁()*+,𝑁)*)/)+,

𝐸)*)/)+,

Change the box size by giving other numbers, e.g., box block 0 6 0 7 0 8.

Change the atom id to delete other atom. Check the lammps document if you want to delete multiple atoms.

0

1

2

3

4

5

6

0 1 2 3 4 5 6

x

y

z

2. Select green FCC atoms1. Coloring with Common neighbor analysis

3. Delete green FCC atoms

View point defect in Ovito

Part II. InterstitialThe formation energy of an interstitial is much larger than that of vacancy because the elastic strain energy of surrounding misfit created by an interstitial atom is substantial.

Interstitial formation energy (IFE)

Octahedral (O) 八面体 Tetrahedral (T) 四面体(¼,¼,¼)(½,½,½) and (½,0,0)

O site added T site added

𝐸1& = 𝐸()*+, −𝑁()*+,𝑁)*)/)+,

𝐸)*)/)+,

1. Create a FCC supercell, 6 x 6 x 6 unit cell dimension, 864 atoms

2. Identify the number of atoms Ninitial in the cell and compute the initial energy Einitial

3. Introduce an additional atom into the supercell to create an interstitial

4. Relax the supercell via energy minimization (conjugate gradient or fire)

5. Identify the number of atoms Nfinal, calculate the final energy Efinal

6. Calculate the interstitial formation energy EIF (units in eV)

Interstitial formation energy (T = 0 K)

dump.0 perfect crystal (864 atoms)

dump.1 add an atom (865 atoms)

dump.2 interstitial atom after minimization (865 atoms)

𝐸1&0 = 𝐸()*+, −𝑁()*+,𝑁)*)/)+,

𝐸)*)/)+,

input fileinterstitial

x,y,z (units in bins)

create a 6x6x6 fcc crystal, units in bins (4 atoms/bin)

Energy minimization

𝐸%&0 = 𝐸()*+, −𝑁()*+,𝑁)*)/)+,

𝐸)*)/)+,

Change the site (x,y,z) to add interstitial atom.

To promote interstitial hopping form one site to the other such that a more energetically favorable site can be found, significant thermal fluctuations may help.

The energy required to overcome the migration energy barrier (Gm) from one atomic site to another is ~ 1 eV/atom. The kinetic or thermal energy of an atom is much smaller, kBT = 0.026 eV at the room temperature. So, thermal fluctuations of large magnitude are necessary for interstitial hopping.

The hopping probability, R, depends on temperature and can be described by Arrhenius equation

where R0 is the attempt frequency proportional to the frequency of atomic vibrations.

𝑅 = 𝑅3exp −𝐺#𝑘8𝑇

G

Δ𝐺#

x

Part III. Thermal Activation of Interstitial

1. Generate a FCC cell with a 6 x 6 x 6 unit cell dimension with 864 atoms

2. Identify the number of atoms Ninitial in the initial supercell with energy Einitial

3. Create an interstitial

4. Heating up the supercell and relax

(1) heating (NVT): increase the temperature from 0.5 K to 300 K

(2) energy minimization (conjugate gradient or fire)

5. Identify the number of atoms Nfinal, calculate the final energy Efinal containing the interstitial

6. Calculate the interstitial formation energy, EIF (eV)

Interstitial Formation Energy via Thermal Activation

𝐸1& = 𝐸()*+, −𝑁()*+,𝑁)*)/)+,

𝐸)*)/)+,

input filethermally activated interstitial

hopping

Change the final temperature, adjustheating rate, try deeper relaxationThe final temperature is 300K

Run with NVT for 5000 timestep

The interstitial site Try different sites, compare to results obtained without thermal activation

gnuplot

gnuplot command

p plotu using1:2 column 1 and 2w withlp line + point$2 value in the second columnq exit

↑ history of the input

Plot data with gnuplot “step : energy : temperature” in data

Dumbbell

Self-interstitials in FCC/BCC crystals

The atomic structure of minimum formation energy

Energy surface

Thermally activated hopping

Octahedral site

Tetrahedral site

Dumbbell structure

Hopping across energy barriers

LevelsAccomplishme

nt of assignments

Quality of writing and

plotsData analyses Creative

learning Independency

A+, A, A- All Excellent Convincing Yes

YesB+, B, B- All Good

Properly but less convincing

Limited

C+, C, C- Less Fair good

F Even less PoorNo analysis or

wrongNo No

(1) Point defects

(2) Dislocations

(3) Equilibrium lattice parameters and bulk moduli (fcc, bcc, hcp, dc)

Submit your reports (1) and (2) in PDF format via emails to cms.sjtu@gmail.com

Email Subject:

Name-StudentID-Point defects.pdfName-StudentID-dislocations.pdf

Combine report (3) with corresponding DFT calculations, submit it to Prof. Kong, along with DFT’s other assignments

Report