Structure and Energy of Stacking Faults - Nithin Thomas

In crystallography, a stacking fault is a type of defect which characterizes the disordering of crystallographic planes. It is thus considered a planar defect. In this work, a brief explanation of the structure and atomistic methods for calculation of generalised Stacking Fault Energy is presented. Two methods using Density Functional Theory namely, Axial Interaction Model (AIM) and Supercell method are discussed.

1. STRUCTURE AND ENERGY OF STACKING FAULTS
NithinThomas, Dr. habil. Thomas Hammerschmidt
[email protected]
14/07/2020 Bochum
© Image reproduced with special permission for non
-commercial use
by Prof. Patrick Cordier FMSA
FAGU
, Institut
Universitaire
de
France, France

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§Stackingfaults
§TypesofStackingFaults
§Stackingfault energy(SFE)
§ImportanceofSFE
§StructureofStackingfaults
§AtomisticmethodsfordeterminationofSFE
§Summary
2Atomisticaspectsofmaterialsproperties| Structureandenergyofstackingfaults
CONTENTS
https://www.materialsdesign.com/datasheet/SQS
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3. STACKING FAULTS
Definition andillustration.

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STACKING FAULTS
:
C
B
A
C
B
A
C
:
§Real materials deviate from ideality.
§Real materials, unlike ideal crystals possess defects or crystal
imperfections controls the physical and mechanical properties of
the material.
§Stacking Fault is a two-dimensional surface defect which is the
fault in the periodic stacking sequence of a crystal.

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STACKING FAULTS
:
C
B
A
C
B
A
C
:
:
A
C
B
A
B
A
C
B
:
+ Fault
HCP
CCP
CCP
Fault Plane
Schematic representation of stacking in F.C.C crystals
ABC

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STACKING FAULTS
A
C
B
A
C
B
A
C
B
:
C
B
A
C
B
A
C
:
B
A
C
B
A
B
A
C
B
§There is no change in orientation of the crystal
across the stacking fault.
§Crystal on one side of SF is shifted by a non-lattice
translation with respect to the crystal on the other
side.

7. TYPES OF STACKING FAULTS
Growth Fault, Deformation Fault andExtrinsicFault

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§A growthfault isformedbyremovingan Aplane abovea B
plane, andthenshearingtheremainingplanes abovetheB
plane by!
"[!"!##], resultingin . . .ABAḂACBCB. . .
Growth Fault
TYPES OF STACKING FAULTS
Effects of Alloying Elements on Stacking Fault Energies and Electronic Structures of Binary Mg Alloys: A First-Principles Study , William Yi Wang et al, Materials Research Letters
Schematic structure of the Mg–X alloys with different stacking faults (shown as (0001)
miller plane), a growth fault
© Schematic
structure of the Mg
–X alloys with different stacking faults (shown as (0001) miller plane), a growth
fault is licensed under CC BY 3.0 by Taylor & Francis

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Deformation Fault
§A deformationfault isformedbya shearingof!
"[!"!##],
resulting in . . .ABȦ%̇&ACA. . .
TYPES OF STACKING FAULTS
Effects of Alloying Elements on Stacking Fault Energies and Electronic Structures of Binary Mg Alloys: A First-Principles Study , William Yi Wang et al, Materials Research Letters
Schematic structure of the Mg–X alloys with different stacking faults (shown as (0001)
miller plane), a deformation fault
© Schematic
structure of the Mg
–X alloys with different stacking faults (shown as (0001) miller plane), a
deformation fault is licensed under CC BY 3.0 by Taylor & Francis

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ExtrinsicFault
§An ExtrinsicFault isgeneratedbyinsertingan extra Cplane
intotheideal HCP structure, resultingin . . .ABÄB̈C̈ABAB. . .
TYPES OF STACKING FAULTS
Effects of Alloying Elements on Stacking Fault Energies and Electronic Structures of Binary Mg Alloys: A First-Principles Study , William Yi Wang et al, Materials Research Letters
Schematic structure of the Mg–X alloys with different stacking faults (shown as (0001)
miller plane), a Extrinsic fault
© Schematic
structure of the Mg
–X alloys with different stacking faults (shown as (0001) miller plane), a
extrinsic fault is licensed under CC BY 3.0 by Taylor & Francis

11. STACKING FAULT ENERGY
Definition andexamples.

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STACKING FAULT ENERGY
§The stacking-fault energy is a materials property on a very
small scale.
§It is noted as γSFEin units of energy per area.
§A stacking fault is an interruption of the normal stacking
sequence of atomic planes in a close-packed crystal
structure. These interruptions carry a certain energy defined
as the stacking-fault energy.
Successive steps for gamma-line calculation. A rigid body shear
is applied (at 30 GPahere) on Mg-Pvalong [010] and in (100).
http://umet.univ-lille1.fr/Projets/RheoMan/en/to-learn-more-about/generalized-stacking-faults.php
© Image reproduced with special permission for non
-commercial use
by Prof. Patrick Cordier FMSA
FAGU
, Institut
Universitaire
de
France, France

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MaterialSFE (mJm-2)
Brass <10
StainlessSteel<10
Silver(Ag) 25
Gold (Au) 75
Silicon (Si) >42
Nickel (Ni) 90
Copper(Cu) 70 -78
Magnesium (Mg) 125
Aluminium (Al) 160 -250
StackingFault Energiesofsomecommonmetalsandalloys
Atomistic aspects of materials properties | Structure and energy of stacking faults
Deformation and Fracture Mechanics of Engineering Materials. John Wiley & Sons, Inc. p.80.Hertzberg, Richard W.; Vinci, Richard P.; Hertzberg, Jason L. (2013).

14. IMPORTANCE OF STACKING FAULT
ENERGY
RelevanceofSFE in deformationofcrystals

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IMPORTANCE OF STACKING FAULT ENERGY
§Stacking Fault Width can be geometrically defined as the
distance between two dislocation partials.
§SF Width determines the mode of deformation.
§Cross-slipis the movement of a screw dislocation from one
allowable slip plane to another.
§SF Width is inversely proportional to SFE.
§When SF Width is low, SFE is high as Cross-slip forms easily and
results in easy deformation.
§A reduction in SFE enhances twinnabilityandductility of the
material and dislocation slip dominates in high SFE materials.
SFE ∝1
SFEWidth∝Cross-slip ∝1
CreepResistance

16. STRUCTURE OF STACKING FAULTS
Structureofdislocationcores

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STRUCTURE OF STACKING FAULTS
§Dislocation core is a region of crystal lattice around the dislocation line in which the
relative displacements of the neighbouring atoms exceed the elastic limit (say 2% in
terms of the local shear strain).
§Peierlsstress is an idealized concept, defined as the minimal stress to move a
dislocation at zero temperature.
§Peierlsbarrier is defined as the energy barrier that a straight dislocation must surmount
in order to move to a neighbouring lattice position -Peierlsvalley.

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STRUCTURE OF STACKING FAULTS
§Much of the dislocation behaviour observed in FCC metals and alloys results from the
Shockley dissociation, by which perfect 1
2⟨110⟩dislocations split into two partial
dislocations, bounding an area of stacking fault.
§In FCC materials stable stacking faults are found only in the {111} planes.
§The three layers are shifted by 1
3⟨111⟩along the plane normal, forming a
repeat pattern with periodicity ⟨111⟩. ABC

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STRUCTURE OF STACKING FAULTS
§A partial dislocation loop can be viewed as the boundary
separating an area of I.S.F from the rest of the plane.
§This partial shift can occur in any of the three equivalent directions
A#$, A#%, A#&
§To make a complete (perfect) dislocation, two atomic layers
bounding the ISF inside the first partial loop is to be shifted again
along another partial shift direction.
§However, to avoid the atoms moving on top of each other, the
second shift should be chosen from a different set of three partial
shift directions.
§Clearly, for every perfect dislocation with Burgers vector A, only
one combination of partial shifts A#$and A#%exists that avoids
atomic run-ons and then only if introduced in a certain order.
!
"!""!#
"!$
'!
'"
'#
Schematic representation of the burgersvectors !!,#,$and partial Burgers vectors !%!,%#,%$on the {111} plane.

20. ATOMISITIC METHODS FOR
DETERMINATION OF SFE
First principlecalculationsofSFE.

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ATOMISITIC METHODS FOR DETERMINATION OF SFE
§Ab-initiototal-energy calculations were based ondensity-functional theoryas
implemented in the Viennaab initiosimulation package (VASP).
§Exchange and correlation interactions are described using a gradient corrected functional
in the Perdew-Burke-Ernzerhof(PBE) form.
§Electron-ion interactions were treated within the Projector Augmented Wave (PAW)
method (From VASP).
§The energy cutofffor theplane-wavebasis set was set to be 270eV unless indicated.
§The energy convergence was set to be 10−6eV.
§All calculations were performed withspin-polarizationto account for themagnetic
propertiesof considered alloys.

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ATOMISITIC METHODS FOR DETERMINATION OF SFE
§The chemical disorder was modelled using Special Quasirandom
Structures (SQS) developed to predict self-averaging quantities of
alloys using finite size supercells.
§SQS structure was constructed by optimization of the Warren-CowleyShort Range Order(SRO) parametersα!"#.
§The optimization of SRO was achieved by swapping elemental
species with a Monte Carlo algorithm.
§The decision whether to accept or to reject the exchange is made
according to the standard Metropolis scheme.
§In this way, the closest random structure at a given supercell size was
generated.

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ATOMISITIC METHODS FOR DETERMINATION OF SFE
Crystal structure of CuAumetal alloy and InGaAs2 semiconductor alloy as generated byMedeASpecial QuasirandomStructures
https://www.materialsdesign.com/datasheet/SQS
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ATOMISITIC METHODS FOR DETERMINATION OF SFE
§Two methods are used for calculation of SFE.
§In the first approach, a parameterized model is obtained by mapping
the stacking sequence onto a one-dimensional Axial Next Nearest
Neighbour Ising(ANNNI) model.
§In this model, theab initio total energy is expressed as a sum of
coupling energies between individual planes that is then truncated to
a finite number of neighbouring layers, under the assumption that the
fault interactions are short ranged.
§The SFE can then be deduced from the calculated coupling
constants.
§This approach can be generalized to the AIM model by including
higher order terms in the expansions.
Edhcp, Ehcpand Efccare the total energy per atom
of thedhcp,hcpandfccphase, respectively

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ATOMISITIC METHODS FOR DETERMINATION OF SFE
§In the supercell approach, an SQS consisting of 9[B1B10] planes
containing 108 atoms was constructed by considering three-
dimensional periodic boundary conditions.
§A vacuum region of more than 10Åwas added and a stacking fault
was inserted by rigidly shifting the upper four[B1B10]layers in the [112]
or[B110] directions.
§This process was repeated for every possible position of the stacking
fault and the stacking sequence was manually changed in order to
ensure that the stacking fault was always located in the middle of the
supercell.
EISFandE0are the energies of configurations
with and without the ISF respectively and A is the
ISF area.

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SUMMARY
§Stacking Fault is a two-dimensional surface which is the fault in the periodic stacking
sequence of a crystal.
§There are different types of Stacking Faults depending on the relative position of the fault
plane.
§SFE ∝1
SFEWidth∝Cross-slip ∝1
CreepResistance
§In FCC materials stable stacking faults are found only in the {111} planes.
§We study the SFEs and γ surfaces for a series of Ni-based CSAs based onfirst-
principlescalculations using both supercell methods and the axial interaction model (AIM)
§In the supercell method, the wholeγsurface is obtained by sliding the upper half of the cell
with respect to the lower half by mapping the stacking sequence into a 1D Isingmodel.

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SUMMARY
§We show that some CSAs exhibit low, even negative SFE at low temperature that
suggestsHCPis more stable thanFCC.
§However, calculation of the temperature dependence of SFE for some CSAs reveals anHCP
to FCCtransition that is driven by the vibrational entropy.
§These results may help to understand the lattice stability and dislocation behaviours in
CSAs.

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FURTHER READING
§Demonstration for SFE calculation using DFT Click here
§First-principles calculations of generalized-stacking-fault-energy of Co-based alloys Click here
§Generalized stacking fault energy in magnesium alloys: Density functional theory calculations Click here
§Stacking fault energies of face-centered cubic concentrated solid solution alloys Click here
§Dislocation Core Effects on Mobility Click here