Comprehensive nuclear materials 1 08 ab initio electronic structure calculations for nuclear materials

Comprehensive nuclear materials 1 08 ab initio electronic structure calculations for nuclear materials Comprehensive nuclear materials 1 08 ab initio electronic structure calculations for nuclear materials Comprehensive nuclear materials 1 08 ab initio electronic structure calculations for nuclear materials Comprehensive nuclear materials 1 08 ab initio electronic structure calculations for nuclear materials Comprehensive nuclear materials 1 08 ab initio electronic structure calculations for nuclear materials Comprehensive nuclear materials 1 08 ab initio electronic structure calculations for nuclear materials

Nuclear Materials

J.-P Crocombette and F Willaime

Commissariat a` l’Energie Atomique, DEN, Service de Recherches de Me´tallurgie Physique, Gif-sur-Yvette, France

ß 2012 Elsevier Ltd All rights reserved.

1.08.4.1.1 Self-interstitials and self-interstitial clusters in Fe and other bcc metals 2321.08.4.1.2 Vacancy and vacancy clusters in Fe and other bcc metals 234

1.08.4.2.1 helium–vacancy clusters in iron and other bcc metals 236

1.08.4.2.3 Interaction of point defects with alloying elements or impurities in iron 2371.08.4.2.4 From dilute to concentrated alloys: the case of Fe–Cr 237

1.08.5.2.1 Bulk electronic structure 243

CTL Charge transition levels

DFT Density functional theory

DLTS Deep level transient spectroscopy

EPR Electron paramagnetic resonance

fcc Face-centered cubic

FLAPW Full potential linearized augmented

plane waves

FP Fission products

GGA Generalized gradient approximation

LDA Local density approximation

LSD Local spin density approximation

LVM Local vibrational modes

PAW Projector augmented waves

PL Photo-luminescence

RPV Reactor pressure vessel

SIA Self-interstitial atom

SQS Special quasi-random structures

TD-DFT Time dependent density functional theory

Electronic structure calculations did not start with

the so-calledab initio calculations or in recent years

The underlying basics date back to the 1930s with an

understanding of the quantum nature of bonding in

solids, the Hartree and Fock approximations, and the

Bloch theorem A lot was understood of the

elec-tronic structure and bonding in nuclear materials

using semiempirical electronic structure calculations,

for example, tight binding calculations.1The

impor-tance of these somewhat historical calculations should

not be overlooked However, in the following sections,

we focus on ‘ab initio’ calculations, that is, density

functional theory (DFT) calculations One must

acknowledge that ‘ab initio calculations’ is a rather

vague expression that may have different meanings

depending on the community In the present chapter

we use it, as most people in the materials science

community do, as a synonym for DFT calculations

The popularity of these methods stems from the factthat, as we shall see, they provide quantitative results

on many properties of solids without any adjustableparameters, though conceptual and technical difficul-ties subsist that should be kept in mind The presen-tation is divided as follows Methodologies and toolsare briefly presented in the first section The nexttwo sections focus on some examples ofab initio results

on metals and alloys on one hand and insulatingmaterials on the other

1.08.2.1 Theoretical Background

In the following a very basic summary of the DFT isgiven The reader is referred to specialized textbooks2–4for further reading and mathematical details Electronicstructure calculations aim primarily at finding theground state of an assembly of interacting nuclei andelectrons, the former being treated classically and thelatter needing a quantum treatment The theoreticalfoundations of DFT were set in the 1960s by theworks of Hohenberg and Kohn They proved that thedetermination of the ground-state wave function ofthe electrons in a system (a function of 3N variables

if the system containsN electrons) can be replaced bythe determination of the ground-state electronic den-sity (a function of only three variables) Kohn and Shamthen introduced a trick in which the density is expressed

as the sum of squared single particle wave functions,these single particles being fictitious noninteractingelectrons In the process, an assembly of interactingelectrons has been replaced by an assembly of fictitiousnoninteracting particles, thus greatly easing the calcu-lations The electronic interactions are gathered in aone-electron term called ‘the exchange and correlationpotential,’ which derives from an exchange and correla-tion functional of the total electronic density Onefinally obtains a set of one-electron Schro¨dinger equa-tions, whose terms depend on the electronic density,thus introducing a self-consistency loop

No exact formulation exists for this exchangeand correlation functional, so one has to resort to

approximations The simplest one is the local density

approximation (LDA) In this approximation, the

den-sity of exchange and correlation energy at a given

point depends only on the value of the electronic

density at this point Different expressions exist for

this dependence, so there are various LDA

func-tionals Another class of functionals pertains to the

generalized gradient approximation (GGA), which

introduces in the exchange and correlation energy

an additional term depending on the local gradient

of the electronic density These two classes of

func-tionals can be referred to as the standard ones Most of

theab initio calculations in materials science are

per-formed with such functionals Recently effort has been

put into the development of a new kind of functional,

the so-called hybrid functionals, which include some

part of exact exchange in their expression Such

func-tionals, which have been used for years in chemistry,

have begun to be used in the nuclear materials

context, though they usually involve much more

time-consuming calculations One of their interests

is that they give a better description of the properties

of insulating materials

We finish this very brief theoretical introduction

by mentioning the concepts ofk-point sampling and

pseudoization

In the community of nuclear materials, most

calculations are done for periodic systems, that is,

one considers a cell periodically repeated in space

Bloch theorem then ensures that the electronic wave

functions should be determined only in the

irreduc-ible Brillouin zone, which is in practice sampled with

a limited number of so-called k points A fine

sam-pling is especially important for metallic systems

Most ab initio calculations use pseudopotentials

Pseudoization is based on the assumption that it is

possible to separate the electronic levels in valence

orbitals and core orbitals Core electrons are

sup-posed to be tightly bound to their nucleus with

their states unaffected by the chemical environment

In contrast, valence electrons fully participate in the

bonding One then first considers in the calculation

that only the valence electrons are modified while the

core electrons are frozen Second, the true interaction

between the valence electrons and the ion made

of the nucleus and core electrons is replaced by a

softer pseudopotential of interaction, which greatly

decreases the calculation burden Various

pseudoiza-tion schemes exist (seeSection 1.08.2.2.2)

Beyond ground-state properties, other theoretical

developments allow the ab initio calculations of

ad-ditional features Detailing these developments is

beyond the scope of this text; let us just mentionamong others time-dependent DFT for electrondynamics, GW calculations for the calculation ofelectronic excitation spectra, density functional per-turbation theory for phonon calculations, and othersecond derivatives of the energy

Ab initio calculations rely on the use of dedicatedcodes Such codes are rather large (a few hundredthousand lines), and their development is a heavytask that usually involves several developers Aneasy, though oversimplified, way to categorize codes

is to classify them in terms of speed on one hand andaccuracy on the other The optimum speed for thedesired accuracy is of course one of the goals of thecode developers (together with the addition of newfeatures) Codes can primarily be distinguished bytheir pseudoization scheme and the type of theirbasis set We will not describe many other numerical

or programming differences, even though they caninfluence the accuracy and speed of the codes.The possible choices in terms of basis sets andpseudoization are discussed in the following para-graphs Pseudoization scheme and basis set are intri-cate as some bases do not need pseudoization andsome pseudoizations presently exist only for specificbasis sets These methodological choices intrinsicallylead to accurate but heavy, or conversely fast butapproximate, calculations We also mention somecodes, though we have no claim to completeness onthat matter Furthermore, we do not comment on theaccuracy and speed of the codes themselves as thedeveloping teams are making continuous efforts toimprove their codes, which make such commentsinappropriate and rapidly outdated

1.08.2.2.1 Basis setsFor what concerns the basis sets we briefly presentplane wave codes, codes with atomic-like localizedbasis sets, and all-electron codes

All-electron codes involve no pseudoizationscheme as all electrons are treated explicitly, thoughnot always on the same footing In these codes, aspatial distinction between spheres close to the nucleiand interstitial regions is introduced Wave functionsare expressed in a rather complex basis set made ofdifferent functions for the spheres and the interstitialregions In the spheres, spherical harmonics asso-ciated with some kind of radial functions (usuallyBessel functions) are used, while in the interstitial

regions wave functions are decomposed in plane

waves All electron codes are very computationally

demanding but provide very accurate results As an

example one can mention the Wien2k5code, which

implements the FLAPW (full potential linearized

augmented plane wave) formalism.6

At the other end of the spectrum are the codes

using localized basis sets The wave functions are

then expressed as combinations of atomic-like

orbi-tals This choice of basis allows the calculations to be

quite fast since the basis set size is quite small

(typi-cally, 10–20 functions per atom) The exact

determi-nation of the correct basis set, however, is a rather

complicated task Indeed, for each occupied valence

orbital one should choose the number of associated

radialz basis functions with possibly an empty

polar-ization orbital The shape of each of these basis

functions should be determined for each atomic

type present in the calculations Such codes usually

involve a norm-conserving scheme for pseudoization

(see the next section) though nothing forbids the use

of more advanced schemes Among this family of

codes, SIESTA7,8 is often used in nuclear material

studies

Finally, many important codes use plane waves as

their basis set.9 This choice is based on the ease of

performing fast Fourier transform between direct and

reciprocal space, which allows rather fast

calcula-tions However, dealing with plane waves means

using pseudopotentials of some kind as plane waves

are inappropriate for describing the fast oscillation of

the wave functions close to the nuclei Thanks to

pseudopotentials, the number of plane waves is

typi-cally reduced to 100 per atom

Finally, we should mention that other basis sets

exist, for instance Gaussians as in the eponymous

chemistry code10 and wavelets in the BigDft

project,11 but their use is at present rather limited

in the nuclear materials community

1.08.2.2.2 Pseudoization schemes

As explained above, pseudoization schemes are

espe-cially relevant for plane wave codes All

pseudoiza-tion schemes are obtained by calculapseudoiza-tions on isolated

atoms or ions The real potential experienced by the

valence electrons is replaced by a pseudopotential

coming from mathematical manipulations A good

pseudopotential should have two apparently

contra-dictory qualities First, it should be soft, meaning that

the wave function oscillations should be smoothened

as much as possible For a plane wave basis set, this

means that the number of plane waves needed to

represent the wave functions is kept minimal Second,

it should be transferable, which means that it shouldcorrectly represent the real interactions of valenceelectrons with the core in any kind of chemical envi-ronment, that is, in any kind of bonding (metallic,covalent, ionic), with all possible ionic charges orcovalent configurations conceivable for the elementunder consideration The generation of pseudopo-tentials is a rather complicated task, but nowadayslibraries of pseudopotentials exist and pseudopoten-tials are freely available for almost any element,though not with all the pseudoization schemes.One can basically distinguish norm-conservingpseudopotentials, ultrasoft pseudopotentials, and PAWformalism Norm-conserving pseudopotentials werethe first ones designed for ab initio calculations.12They involve the replacement of the real valencewave function by a smooth wave function of equalnorm, hence their name Such pseudopotentials arerather easy to generate, and several libraries existwith all elements of the periodic table They are rea-sonably accurate although they are still rather hard,and so they are less and less used in plane wave codesbut are still used with atomic-like basis sets Ultrasoftpseudopotentials13 remove the constraint of normequality between the real and pseudowave functions.They are thus much softer though less easy to gen-erate than norm-conserving ones The ProjectorAugmented Wave14formalism is a complex pseudoi-zation scheme close in spirit to the ultrasoft scheme but

it allows the reconstruction of the real electronic sity and the real wave functions with all their oscilla-tions, and for this reason this method can be considered

den-an all-electron method When correctly generated,PAW atomic data are very soft and quite transferable.Libraries of ultrasoft pseudopotentials or PAW atomicdata exist, but they are generally either incomplete ornot freely available

Plane wave codes in use in the nuclear materialscommunity include VASP15with ultrasoft pseudopo-tentials and PAW formalism, Quantum-Espresso16with norm-conserving and ultrasoft pseudopotentialsand PAW formalism, and ABINIT17 with norm-conserving pseudopotentials and PAW formalism.Note that for a specific pseudoization schememany different pseudopotentials can exist for agiven element Even if they were built using thesame valence orbitals, pseudopotentials can differ bymany numerical choices (e.g., the various matchingradii) that enter the pseudoization process

We present in the following a series of practicalchoices to be made when one wants to perform

ab initio calculations But the first and certainly most

important of these choices is that of the ab initio

code itself as different codes have different speeds,

accuracies, numerical methods, features, input files,

and so on, and so it proves quite difficult to change

codes in the middle of a study Furthermore, one

observes that most people are reluctant to change

their usual code as the investment required to fully

master the use of a code is far from negligible (not to

mention the one to master what isin the code)

1.08.2.3 Ab Initio Calculations in Practice

In this paragraph, we try to give some indication of

what can be done with anab initio code and how it is

done in practice The calculation starts with the

posi-tioning of atoms of given types in a calculation cell of

a certain shape That would be all if the calculations

were trulyab initio Unfortunately, a few more pieces

of information should be passed to the code; the

most important ones are described in the final section

The first section introduces the basic outputs of

the code, and the second one deals with the possible

cell sizes and the associated CPU times

1.08.2.3.1 Output

We describe in this section the output of ab initio

calculations in general terms The possible

applica-tions in the nuclear materials field are given below

The basic output of a standardab initio calculation is

the complete description of the electronic ground

state for the considered atomic configuration From

this, one can extract electronic as well as energetic

information

On the electronic side, one has access to the

elec-tronic density of states, which will indicate whether

the material is metallic, semiconducting, or insulating

(or at least what the code predicts it to be), its

possi-ble magnetic structure, and so on Additional

calcula-tions are able to provide additional information on

the electronic excitation spectra: optical absorption,

X-ray spectra, and so on

On the energetic side, the main output is the total

energy of the system for the given atomic

configura-tion Most codes are also able to calculate the forces

acting on the ions as well as the stress tensor acting on

the cell Knowing these forces and stress, it is possible

to chain ground-state calculations to perform various

spec-
Starting from two relaxed configurations close inspace, one can calculate the energetic path in spacejoining these two configurations, thus allowing thecalculation of saddle points

The integration of the forces in a MolecularDynamics scheme leads to so-called ab initiomolecular dynamics (see Chapter1.09, Molecu-lar Dynamics) Car–Parrinello molecular dynam-ics18 calculations, which pertain to this class ofcalculations, introduce fictitious dynamics onthe electrons to solve the minimization problem

on the electrons simultaneously with the realion dynamics

1.08.2.3.2 Cell sizes and correspondingCPU times

The calculation time ofab initio calculations varies –

to first order – as the cube of the number of atoms orequivalently of electrons (the famousN3

dependence)

in the cell If a finek-point sampling is needed, thisdependence is reduced to N2

as the number of kpoints decreases in inverse proportion with the size

of the cell On the other hand, the number of consistent cycles needed to reach convergence tends

self-to increase withN Anyway, the variation of tion time with the size of the cell is huge and thusstrongly limits the number of atoms and also the cellsize that can be considered On one hand, calculations

calcula-on the unit cell of simple crystalline materials (with asmall number of atoms per unit cell) are fast and caneasily be performed on a common laptop On theother hand, when larger simulation cells are needed,the calculations quickly become more demanding.The present upper limit in the number of atomsthat can be considered is of the order of a fewhundreds The exact limit of course depends on thecode and also on the number of electrons per atomsand other technicalities (number of basis functions,kpoints, available computer power, etc.), so it is notpossible to state it precisely Considering such largecells leads anyway to very heavy calculations inwhich the use of parallel versions of the codes is

almost mandatory Various parallelization schemes

are possible: on k points, fast Fourier transform,

bands, spins; the parallelization schemes actually

available depend on the code

The situation gets even worse when one notes that

a relaxation roughly involves at least ten ground-state

calculations, a saddle point calculation needs about

ten complete relaxations, and that each molecular

dynamics simulation time step (of about 1 fs) needs

a complete ground-state calculation Overall, one can

understand that the CPU time needed to complete an

ab initio study (which most of the time involves

vari-ous starting geometry) may amount up to hundreds

of thousands or millions of CPU hours

1.08.2.3.3 Choices to make

Whatever the system considered and the code used,

one needs to provide more inputs than just the atomic

positions and types Most codes suggest some values

for these inputs However, their tuning may still be

necessary as default values may very well be suited

for some supposedly standard situations and

irrele-vant for others Blind use ofab initio codes may thus

lead to disappointing errors Indeed, not all these

choices are trivial, so mistakes can be hard to notice

for the beginner Choices are usually made out of

experience, after considering some test cases needing

small calculation time

One can distinguish between choices that should

be made only once at the beginning of a study and

calculation parameters that can be tuned calculation

by calculation The main unchangeable choices are

the exchange and correlation functional and the

pseudopotentials or PAW atomic data for the various

atomic types in the calculation

First, one has to choose the flavor of the exchange

and correlation functional that will be used to

describe the electronic interactions Most of the

time one chooses either an LDA or a GGA

func-tional Trends are known about the behavior of these

functionals: LDA calculations tend to overestimate

the bonding and underestimate the bond length in

bulk materials, the opposite for GGA However,

things can become tricky when one deals with defects

as energy differences (between defect-containing and

defect-free cells) are involved For insulating

materi-als or materimateri-als with correlated electrons, the choice

of the exchange and correlation functional is even

more difficult (seeSection 1.08.5)

The second and more definitive choice is the one

of the pseudopotential We do not mean here the

choice of the pseudoization scheme but the choice

of the pseudopotential itself Indeed, calculated gies vary greatly with the chosen pseudopotential, soenergy differences that are thermodynamically orkinetically relevant are meaningless if the variouscalculations are performed with different pseudopo-tentials The determination of the shape of the atomicbasis set in the case of localized bases is also ofimportance, and it is close in spirit to the choice ofthe pseudopotential except that much less basis setsthan pseudopotentials are available

ener-More technical inputs include

the k-point sampling The larger the number of

k points to sample the Brillouin zone, the moreaccurate the results but the heavier the calcula-tions will be This is especially true for metallicsystems that need fine sampling of the Brillouinzone, but convergence with respect to the number

ofk points can be accelerated by the introduction

of a smearing of the occupations of electroniclevels close to the Fermi energy The shape andwidth of this smearing function is then an addi-tional parameter.19

the number of plane waves (obviously for plane wavecodes but also for some other codes that also useFFT) Once again the larger the number of planewaves, the more accurate and heavier the calculation

the convergence criteria The two major gence criteria are the one for the self-consistentloop of the calculation of the ground-state electronicwave functions and the one to signal the convergence

conver-of a relaxation calculation (with some thresholddepending on the forces acting on the atoms)

1.08.3 Fields of Application

Ab initio calculations can be applied to almost anysolid once the limitations in cell sizes and number ofatoms are taken into account Among the materials

of nuclear interest that have been studied one can citethe following: metals, particularly iron, tungsten,zirconium, and plutonium; alloys, especially ironalloys (FeCr, FeC to tackle steel, etc.); models offuel materials, UO2, U–PuO2, and uranium carbides;structural carbides (SiC, TiC, B4C, etc.); waste mate-rials (zircon, pyrochlores, apatites, etc.)

In this section, we rapidly expose the types ofstudies that can be done with ab initio calculations.The last two sections on metallic alloys and insulat-ing materials will allow us to go into detail for somespecific cases

1.08.3.1 Perfect Crystal

1.08.3.1.1 Bulk properties

Dealing with perfect crystals, ab initio calculations

provide information about the crystallographic and

electronic structure of the perfect material The

properties of usual materials, such as standard metals,

band insulators, or semi-conductors, are basically

well reproduced, though some problems remain,

es-pecially for nonconductors (seeSection 1.08.5.1 on

SiC) However, difficulties arise when one wishes to

tackle the properties of highly correlated materials

such as uranium oxide (Section 1.08.5.2) For

in-stance, no ab initio code, whatever the complexity

and refinements, is able to correctly predict the fact

that plutonium isnonmagnetic In such situations, the

nature of the chemical bonding is still poorly

under-stood, so the correct physical ingredients are

proba-bly not present in today’s codes These especially

difficult cases should not mask the very impressive

precision of the results obtained for the crystal

struc-ture, cohesive energy, atomic vibrations, and so on of

less difficult materials

1.08.3.1.2 Input for thermodynamic models

The information on bulk materials can be gathered in

thermodynamical models Mostab initio calculations

are performed at zero temperature Even with this

restriction, they can be used for thermodynamical

studies First, ab initio calculations enable one to

consider phases that are not accessible to

experi-ments It is thus possible to compare the relative

stability of various (real or fictitious) structures for

a given composition and pressure

Considering alloys, it is possible to calculate the

cohesive energy of various crystallographic

arrange-ments Solid solutions can also be modeled by so-called

special quasi-random structures (SQS).20 Beyond a

simple comparison of the energies of the various

struc-tures, when a common underlying crystalline network

exists for all the considered phases, the information

about the cohesive energies can be used to

parameter-ize rigid lattice inter-atomic interaction models (i.e.,

pair, triplet, etc., interactions) that can be used to

per-form computational thermodynamics (see Chapter

1.17, Computational Thermodynamics: Application

to Nuclear Materials) These interactions can then be

used in mean field or Monte-Carlo simulations to

predict phase stabilities at nonzero temperature.21

As examples of this kind of studies one can cite the

determination of solubility limits (e.g., Zr and Sc in

aluminum22) and the exploration of details of the

phase diagrams (e.g., the inversion of stability in theiron-rich side of the Fe–Cr diagram23)

Directly considering nonzero temperature in

ab initio simulation is also possible, though moredifficult First, one can calculate for a given composi-tion and structure the electronic and vibrationalentropy (through the phonon spectrum), which leads

to the variation in heat capacity with temperature.Nontrivial thermodynamic integrations can then beused to calculate the relative stability of various struc-tures at nonzero temperature Second, one can perform

ab initio molecular dynamics simulations to modelfinite temperature properties (e.g., thermal expansion).1.08.3.2 Defects

Point defects are of course very important in a nuclearcomplex as they are created either by irradiation or

by accommodation of impurities (e.g., fission ucts (FP)) (see Chapter 1.02, Fundamental Point

Radiation-Induced Effects on Microstructure).More generally, they have a tremendous role in thekinetic properties of the materials It is therefore notsurprising if countless ab initio studies exist on pointdefects in nuclear materials Most of them are based

on a supercell approach in which the unit cell of theperfect crystal is periodically repeated up to the larg-est possible simulation box A point defect is thenintroduced, and the structure is allowed to relax Bydifference with the defect-free structure, one can cal-culate the formation energy of the defect that drives itsequilibrium concentration Some care must be taken inwriting this difference as the number and types ofatoms should be preserved in the process Point defectsare also the perfect object for the saddle point calcula-tions that give the energy that drives their kineticproperties Ab initio permits accurate calculation ofthese energies and also consideration of (for insulatingmaterials) the various possible charge states of thedefects They have shown that the properties of defectscan vary greatly with their charge states

Many different kinds of defects can be considered

A list of possible defects follows with the characteristicassociated thermodynamical and kinetic energies.1.08.3.2.1 Self-defects

Vacancies and interstitials, with the associated tion energy driving their concentration and migra-tion energy driving their displacement in the solid;the sum of these two energies is the activation energyfor diffusion at equilibrium For such simple defects,

forma-it is possible to go beyond the 0 K energies and to

access the free energies of formation and migrations

by calculating the vibrational spectra in the presence

of the defect in the stable position and at the saddle

point (seeSection 1.08.4.2.3)

1.08.3.2.2 Hetero-defects

In the nuclear context, such defects can be fission

products in a fuel material, actinide atoms in a waste

material, helium gases in structural materials, and

so on;ab initio gives access to the solution energy of

these impurities, which allows one to determine their

most favored positions in the crystal: interstitial

position, substitution for host atoms, and so on The

kinetic energies of migration of interstitial impurities

are accessible as well as the kinetic barrier for the

extraction of an impurity from a vacancy site

1.08.3.2.3 Point defect assemblies

In this class, one can include the calculation of

inter-stitial assemblies as well as the complexes built with

impurities and vacancies One then has access to the

binding of monoatomic defects to the complexes,24

possibly with the associated kinetic energy barriers

1.08.3.2.4 Kinetic models

As for perfect crystals, the information obtained by

ab initio calculations can be gathered and integrated in

larger scale modeling, especially, kinetic models Many

kinetic Monte-Carlo models were thus parameterized

withab initio calculations (see e.g., the works on pure

iron25 or FeCu26and Chapter 1.14, Kinetic Monte

Carlo Simulations of Irradiation Effects)

1.08.3.2.5 Extended defects

Even if the cell sizes accessible byab initio calculations

are small, it is possible to deal with some extended

defects Calculations then often need some tricks to

accommodate the extended defect in the small cells

Some examples are given in the next section on

stud-ies on dislocations

1.08.3.3 Ab Initio for Irradiation

Irradiation damage, especially cascade modeling, is

usually preferentially dealt by larger scale methods

such as molecular dynamics with empirical potentials

rather than ab initio calculations However, recently

ab initio studies that directly tackle irradiation

pro-cesses have appeared

1.08.3.3.1 Threshold displacement energies

First, the increase in computer power has allowed the

calculations of threshold displacement energies by

ab initio molecular dynamics We are aware of studies

in GaN27and silicon carbides.28,29The procedure isthe same as that with empirical potentials: one initi-ates a series of cascades of low but increasing energyand follows the displacement of the acceleratedatom The threshold energy is reached as soon asthe atom does not return to its initial position atthe end of the cascade Such calculations are verypromising as empirical potentials are usually im-precise for the orders of energies and interatomicdistances at stake in threshold energies However,they should be done with care as most pseudopo-tentials and basis sets are designed to work formoderate interatomic distances, and bringing twoatoms too close to each other may lead to spuriousresults unless the pseudopotentials are specificallydesigned

1.08.3.3.2 Electronic stopping powerSecond, recent studies have been published in the

ab initio calculations of the electronic stopping powerfor high-velocity atoms or ions The frameworkbest suited to address this issue is time-dependentDFT (TD-DFT) Two kinds of TD-DFT have beenapplied to stopping power studies so far

The first approach relies on the linear response

of the system to the charged particle The key tity here is the density–density response functionthat measures how the electronic density of the solidreacts to a change in the external charge density.This observable is usually represented in reciprocalspace and frequency, so it can be confronted directlywith energy loss measurements The density–density response function describes the possibleexcitations of the solid that channel an energy trans-fer from the irradiating particle to the solid Mostnoticeably the (imaginary part of the) functionvanishes for an energy lower than the band gapand shows a peak around the plasma frequency.Integrating this function over momentum andenergy transfers, one obtains the electronic stoppingpower Campillo, Pitarke, Eguiluz, and Garcia haveimplemented this approach and applied to somesimple solids, such as aluminum or silicon.30–32They showed that there is little difference betweenthe usual approximations of TD-DFT: the randomphase approximation, which means basically noexchange correlation included, or adiabatic LDA,which means that the exchange correlation is local

quan-in space and quan-instantaneous quan-in time The quan-influence

of the band structure of the solid accounts fornoticeable deviations from the homogeneous elec-tron gas model

The second approach is more straightforward

conceptually but more cumbersome technically It

proposes to simply monitor the slowing down of the

charged irradiated particle in a large box in real space

and real time The response of the solid is hence not

limited to the linear response: all orders are

automat-ically included However, the drawback is the size of

the simulation box, which should be large enough to

prevent interaction between the periodic images

Fol-lowing this approach, Pruneda and coworkers33

cal-culated the stopping power in a large band gap

insulator, lithium fluoride, for small velocities of the

impinging particle In the small velocity regime, the

nonlinear terms in the response are shown to be

important

Unfortunately, whatever the implementation of

TD-DFT in use, the calculations always rely on very

crude approximations for the exchange-correlation

effects The true exchange-correlation kernel

(the second derivative of the exchange-correlation

energy with respect to the density) is in principle

nonlocal (it is indeed long ranged) and has memory

The use of novel approximations of the kernel was

recently introduced by Barriga-Carrasco but for

homogeneous electron gas only.34,35

1.08.3.4 Ab Initio and Empirical Potentials

Ab initio calculations are often compared to and

sometimes confused with empirical potential

calcu-lations We will now try to clarify the differences

between these two approaches and highlight their

point of contacts The main difference is of course

thatab initio calculations deal with atomic and

elec-tronic degrees of freedom Empirical potentials

depend only on the relative positions of the

consid-ered atoms and ions They do not explicitly consider

electrons Thus, roughly speaking, ab initio

calcula-tions deal with electronic structure and give access to

good energetics, whereas empirical potentials are not

concerned with electrons and give approximate

energetics but allow much larger scale calculations

(in space and time)

Going into some details, we have shown that

ab initio gives access to very diverse phenomena

Some can be modeled with empirical potentials,

at least partly; others are completely outside the

scope of such potentials

In the latter category, one will find the phenomena

that are really related to the electronic structure

itself For instance, the calculations of electronic

excitations (e.g., optical or X-ray spectra) are

concep-tually impossible with empirical potentials In the

same way, for insulating materials, the calculation ofthe relative stability of various charge states of a givendefect is impossible with empirical potentials.Other phenomena that are intrinsically electronic

in nature can be very crudely accounted for in ical potentials The electronic stopping power of

empir-an accelerated particle is empir-an example As indicatedabove, it can be calculatedab initio Conversely, fromthe empirical potential perspective one can add an

ad hoc slowing term to the dynamics of fast ing particles in solids whose intensity has to be estab-lished by fitting experimental (or ab initio) data

mov-In a related way, some forms of empirical potentialsrely on electronic information; for instance, theFinnis–Sinclair36 or Rosato et al.37

forms In thesame spirit, a recent empirical potential has beendesigned to reproduce the local ferromagnetic order

of iron.38However, this potential assumes a tendencyfor ferromagnetic order, whileab initio calculation can(in principle) predict what the magnetic order will be.Therefore,ab initio is very often used as a way toget accurate energies for a given atomic arrangement.This is the case for the formation and migrationenergies of defects, the vibration spectra, and so on.These phenomena are conceptually within reach ofempirical potentials (except the ones that reincorpo-rate electronic degrees of freedom such as chargeddefects).Ab initio is then just a way to get proper andquantitative energetics Their results are often used

as reference for fitting empirical potentials However,the fit of a correct empirical remains a tremendoustask especially with the complex forms of potentialsnowadays and when one wants to correctly predictsubtle, out of equilibrium, properties

Finally, one should always keep in mind thatcohesion in solids is quantum in nature, so classicalinteratomic potentials dealing only with atoms orions can never fully reproduce all the aspects ofbonding in a material

The vast majority of DFT calculations on radiationdefects in metallic materials have been performed inbody-centered cubic (bcc) iron-based materials, forobvious application reasons of ferritic steels but alsobecause of the more severe shortcoming of predic-tions based only on empirical potentials A number

of accurate estimates of energies of formation andmigration of self-interstitial and vacancy defects

as well as small defect clusters and solute-vacancy

or solute-interstitial complexes have been obtained

DFT calculations have been intensively used to

predict atomistic defect configurations and also

tran-sition pathways An overview of these results is

pre-sented below, complete with examples in other bcc

transition metals, in particular tungsten, as well as

hcp-Zr These examples illustrate how DFT data

have changed the more or less admitted energy

land-scape of these defects and also how they are used to

improve empirical potentials In the final part of this

chapter, a brief overview of typical works on

disloca-tions (in iron) is presented

1.08.4.1 Pure Iron and Other bcc Metals

Ferritic steels are an important class of nuclear

mate-rials, which include reactor pressure vessel (RPV)

steels and high chromium steels for elevated

temper-ature structural and cladding materials in fast reactors

and fusion reactors, seeChapter4.03, Ferritic Steels

and Advanced Ferritic–Martensitic Steels From a

basic science point of view, the modeling of these

materials starts with that of pure iron, in the

ferro-magnetic bcc structure Iron presents several

difficul-ties for DFT calculations First, being a three

dimensional (3D) metal, it requires rather large basis

sets in plane wave calculations Second, the calculations

need to be spin polarized, to account for magnetism,

and this at least doubles the calculation time But most

of all, it is a case where the choice of the

exchange-correlation functional has a dramatic effect on bulk

properties The standard LDA incorrectly predicts

the paramagnetic face-centered cubic (fcc) structure

to be more stable than the ferromagnetic bcc structure

The correct ground state is recovered using gradient

corrected functionals,39 as illustrated in Figure 1

Finally, it was pointed out that pseudopotentials tend

to overestimate the magnetic energy in iron,40 and

therefore, some pseudopotentials suffer from a lack of

transferability for some properties In practice,

how-ever, in the large set of the results obtained over the last

decade for defect calculations in iron, a quite

remark-able agreement is obtained between the various

computational approaches With a few exceptions,

they are indeed quite independent on the form of the

GGA functional, the basis set (plane wave or localized),

and the pseudopotential or the use of PAWapproaches

1.08.4.1.1 Self-interstitials and

self-interstitial clusters in Fe and other bcc metals

The structure and migration mechanism of

self-interstitials in iron is a very good illustrative example

of the impact of DFT calculations on radiation defect

studies Progress in methods, codes, and computerperformance made this archetype of radiationdefects accessible to DFT calculations in the early2000s, since total energy differences between simu-lation cells of 128þ1 atoms could then be obtainedwith a sufficient accuracy In 2001, Domain andBecquart reported that, in agreement with theexperiment, theh110i dumbbell was the most stablestructure.41Quite unexpectedly, theh111i dumbbellwas predicted to be 0.7 eV higher in energy, atvariance with empirical potential results that pre-dicted a much smaller energy difference DFT cal-culations performed in other bcc metals revealedthat this is a peculiarity of Fe,42 as illustrated

the origin of the energy increase in theh111i bell in Fe The important consequence of this result

dumb-in Fe, which has been confirmed repeatedly sdumb-ince

20 15 10 5

0 2.40 2.45 2.50 2.55 2.60

s arbitrary units (a.u.)

P-fcc (LSD)

P-bcc (LSD)

P-bcc (PW) Fe

P-fcc (PW)

F-bcc (PW)

F-bcc (LSD)

2.65 2.70 2.75

55 50 45 40 35 30 25

Figure 1 Calculated total energy of paramagnetic (P) bcc and fcc and ferromagnetic (F) bcc iron as a function of Wigner–Seitz radius (s) The dotted curve corresponds to the local spin density (LSD) approximation, and the solid curve corresponds to the GGA functional proposed by Perdew and Wang in 1986 (PW) The curves are displaced

in energy so that the minima for F bcc coincide Energies are in Ry (1 Ry ¼ 13.6057 eV) and distances in bohr (1 bohr ¼ 0.5292 A˚) Reproduced from Derlet, P M.; Dudarev, S L Prog Mater Sci 2007, 52, 299–318.

then, is that it excludes the SIA migration to

occur by long 1D glides of the h111i dumbbell

followed by on-site rotations of theh110i dumbbell,

as predicted previously from empirical potential

MD simulations Moreover, DFT investigation of

the migration mechanism yielded a quantitative

agreement with the experiment for the energy

of the Johnson translation–rotation mechanism (see

Figure 3), namely0.3 eV.43

These DFT calculations were followed by a very

successful example of synergy between DFT and

empirical potentials The DFT values of interstitial

formation energies in various configurations and

interatomic forces in a liquid model have indeed

been included in the database for a fit of EAM type

potentials by Mendelev et al.45

This approach hasresulted in a new generation of improved empirical

potentials, albeit still with some limitations When

considering SIA clusters made of parallel dumbbells,

the Mendelev potential agrees with DFT for

predict-ing a crossover as a function of cluster size from the

h110i to the h111i orientation between 4 and 6 SIA

clusters.44 However, discrepancies are found whenconsidering nonparallel configurations.46 More pre-cisely, new configurations of small SIA clusters wereobserved in MD simulations performed at high tem-perature with the Mendelev potential The energy

of the new di-interstitial cluster, made of a triangle ofatoms sharing one site (see Figure 4), is even lowerthan that of the parallel configuration within DFTbut higher by 0.3 eV with the Mendelev potential (seealso Section 1.08.4.3 on dislocations) The new tri-and quadri-interstitial clusters, with a ring structure(seeFigure 4), are one of the few examples in which asignificant discrepancy is found between variousDFT approaches Calculations with the most accu-rate description of the ionic cores predict that thenew tri-interstitial configuration is slightly more sta-ble than the parallel configuration, whereas moreapproximate ones predict that it is 0.7 eV higher.The first category includes calculations in the PAWapproach, performed using either the VASP code orthe PWSCF code and also ultrasoft pseudopotentialcalculations The second one includes calculations

0.5

1.5

V Nb Ta

W Mo Cr 2.5

Figure 2 Formation energies of several basic SIA configurations calculated for bcc transition metals of group 5B (left) and group 6B (right), taken from Nguyen-Manh et al 42 Data for bcc Fe are taken from Fu et al 43 Reproduced from Nguyen-Manh, D.; Horsfiels, A P.; Dudarev, S L Phys Rev B 2006, 73, 020101.

with less transferable ultrasoft pseudopotentials with

VASP and norm-conserving pseudopotentials with

SIESTA.46 Such a discrepancy is not common in

defect calculations in metals Further investigations

are required to understand more precisely its origin,

in particular the possible role of magnetism

The structures of the most stable SIA clusters in

Fe, and more generally of their energy landscape,

remain an open question One would ideally need

to combine DFT calculations with methods for

ex-ploring the energy surface, such as the Dimer47 or

ART48 methods Such a combination is possible in

principle, and it has indeed been used for defects in

semiconductors,49but due to computer limitations this

is not the case yet in Fe The alternative is to develop

new empirical potentials in better agreement with

DFT energies in particular for these new structures,

to perform the Dimer or ART calculations with thesepotentials, and to validate the main features of theenergy landscape thus obtained by DFT calculations

To summarize, the energy landscape of interstitialtype defects has been revisited in the last decadedriven by DFT calculations, in synergy with empiri-cal potential calculations

1.08.4.1.2 Vacancy and vacancy clusters

in Fe and other bcc metalsDFT has some limitations in predicting accuratevacancy formation energies in transition metals Theexceptional agreement with the experiment obtainedinitially within DFT-LDA50was later shown to resultfrom a cancellation between two effects First, the

Figure 4 New low-energy configurations of SIA clusters in Fe, which revealed discrepancies between DFT and

empirical potentials and between various approximations within DFT Reproduced from Terentyev, D A.; Klaver, T P C.; Olsson, P.; Marinica, M C.; Willaime, F.; Domain, C.; Malerba, L Phys Rev Lett 2008, 100, 145503.

[110]

[111] Crowd [111] [011]

0.0 0.2 0.4 0.6

0.8

DFT-GGA Mendelev Ackland

Figure 3 Left: Johnson translation–rotation mechanism of the h110i dumbbell; white and black spheres indicate the initial and final positions of the atoms, respectively Reproduced from Fu, C C.; Willaime, F.; Ordejon, P Phys Rev Lett 2004,

92, 175503 Right: Comparison between the DFT-GGA result and two EAM potentials for the energy barriers of the Johnson mechanism and the h110i to h111i transformation Reproduced from Willaime, F.; Fu, C C.; Marinica, M C.; Torre, J D.; Nucl Instrum Meth Phys Res B 2005, 228, 92.

structural relaxation, which was neglected by Korhonen

et al.50

is now known to significantly reduce the vacancy

formation energy, in particular in bcc metals.51Second,

due to limitations of exchange-correlation functionals

at surfaces, DFT-LDA tends to underestimate the

vacancy formation energy This discrepancy is even

larger within DFT-GGA, and it increases with the

number of valence electrons It is therefore rather

small for early transition metals (Ti, Zr, Hf,), but it is

estimated to be as large as 0.2 eV in LDA and 0.5 eV in

GGA-PW1 for late transition metals (Ni, Pd, Pt).52

However, the effect is much weaker for migration

ener-gies.52A new functional, AM05, has been proposed to

cope with this limitation.53

Less spectacular effects are expected in

vacancy-type defects than in interstitial-vacancy-type defects when

going from empirical potentials to DFT calculations

The discussion on vacancy-type defects in Fe will be

restricted to the results obtained within DFT-GGA,

due to the superiority of this functional for bulk

prop-erties For pure Fe, DFT-GGA vacancy formation and

migration energies are in the range of 1.93–2.23 eVand

0.59–0.71 eV.41,43,54 These values are in agreement

with experimental estimates at low temperatures in

ultrapure iron, namely 2.0 0.2 eV and 0 55 eV,

respectively These values can be reproduced by

empirical potentials when included in the fit, but one

discrepancy remains with DFT concerning the shape

of the migration barrier It is indeed clearly a single

hump in DFT25 and usually a double hump with

empirical potentials

Concerning vacancy clusters, the structures

pre-dicted by empirical potentials, namely compact

struc-tures, were confirmed by DFT calculations, but

there are discrepancies in the migration energies In

both cases, the most stable divacancy is the

next-nearest-neighbor configuration, with a binding

en-ergy of 0.2–0.3 eV.25,55,56The migration can occur by

two different two-step processes, with an

intermedi-ate configuration that is either nearest neighbor or

fourth nearest neighbor.56A quite unexpected result

of DFT calculations was the prediction of rather low

migration energies for the tri- and quadrivacancies,

namely 0.35 and 0.48 eV.25Depending on the

poten-tial, this phenomenon is either not reproduced or only

partly reproduced (seeFigure 5).57

Stronger deviations from empirical potential

pre-dictions for divacancies are observed in DFT

calcula-tions performed in other bcc metals The most

dramatic case is that of tungsten, where the

next-nearest-neighbor interaction is strongly repulsive

(0.5 eV) and the nearest-neighbor interaction is

vanishing.58 This result does not explain why voidsare formed in tungsten under irradiation

1.08.4.1.3 Finite temperature effects ondefect energetics

The properties of radiation defects at high ture may change due to three possible contributions tothe free energy: electronic, magnetic, and vibrational.These three effects can be well modeled in bulk bcciron,59but they are more challenging for defects Theelectronic contribution, which exists only in metals,arises due to changes in the density of states close tothe Fermi level The electronic entropy differencebetween, for example, two configurations is, to firstorder, proportional to the temperature, T, and thechange in density of states at the Fermi level Thiselectronic effect is straightforward to take into account

tempera-in DFT calculations It was shown tempera-in tungsten todecrease the activation free energy for self-diffusion

by up to 0.4 eV close to the melting temperature Thus,although this effect is relatively small in general, itcannot be neglected at high temperature

The magnetic contribution is important in iron.Spin fluctuations were shown to be the origin of thestrong softening of the C0elastic constant observed asthe temperature increases up to the ag transitiontemperature,60and it drives, for instance, the tempera-ture dependence of relative abundance of<100> and

<111> interstitial loops formed under irradiation.61

It is also known to have a small effect on vacancyproperties, but to the authors’ knowledge there is pres-ently no tractable method to predict this effect forpoint defects quantitatively from DFT calculations.This is probably one of the important challenges inthe field

0.4 0.6 0.8 1.0

1.2 Ackland et al. Mendelev et al.9445

Figure 5 Migration energies of vacancy clusters in Fe,

as a function of cluster size Reproduced from Fu, C C.; Willaime, F (2004) Unpublished.