Materials Modelling using Density Functional Theory Properties and Predictions

303 Pages·2014·23.152 MB·English

by  Feliciano Giustino

Materials Modelling using Density Functional Theory

Materials Modelling using Density
Functional Theory
Properties and Predictions
Feliciano Giustino
Department of Materials, University of Oxford
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To Nicola and Anna
‘For whatever it’s worth, I’m here to tell you that it is possible. It is possible.’
— V. A. Freeman

Preface
This book is intended to be an introduction to the modelling of materials starting
from the first principles of quantum mechanics. For the reader who is not familiar
withthenotionof‘first-principlescalculations’,onecouldsimplysaythatthisbookis
about the art of using the periodic table, quantum mechanics and computers in order
to understand and possibly predict certain properties of materials.
The primary audience for this book are senior undergraduate and first-year graduate
students in Materials Science, Physics, Chemistry and Engineering. Experienced
researchers and engineers who are approaching the quantum theory of materials for
the first time may also find this book useful as a light-touch introduction.
As advanced materials make their way into every aspect of modern life and society,
from electronics to construction, transport and energy, the use of quantum mechanics
inthequantitativenon-empiricalmodellingofmaterialsisexperiencingarapidgrowth
in university departments, research institutes and industrial laboratories alike.
Following this trend, in the Department of Materials at the University of Oxford we
felt the need to complement the undergraduate curriculum in Materials Science with
introductorycoursesonmaterialsmodellingusingquantummechanics.Thisbookwas
inspiredbyaseriesoflecturesdeliveredatOxfordbytheauthor,fortwoundergraduate
modules in Materials Science: Electronic structure of materials (second year) and
Prediction of material properties using density functional theory (third year). The
lattercourseisalsoattendedbydoctoralstudentsintheirfirstyear.Thepresentbook
was written to make the learning of atomistic materials modelling as easy as possible,
in anticipation that this subject will become a core discipline in Materials Science
education.
The key idea of this book is to introduce the reader to advanced concepts in quantum
mechanics and materials science by assuming only elementary prior knowledge of
classicalmechanics,electrodynamicsandquantummechanics,atthelevelexpectedfor
students in Materials Science or Engineering. Students reading Physics or Chemistry
will be already familiar with many of the concepts presented in this book, and will
mostly benefit from the concise and pragmatic introduction to density functional
theory and its uses.
At the time of writing, the many textbooks available on materials modelling broadly
fall into two categories. On the one hand we find books devoted to the theory of
molecules and solids, including several classics which shaped entire generations of
scientists, such as Kittel (1976) and Ashcroft and Mermin (1976) for solids. These
books generally pre-dated the rise of density functional theory; hence they inevitably
miss those aspects of materials modelling relating to ‘first-principles calculations’.
On the other hand we have presentations devoted to density functional theory and
its applications, which are widely adopted and have become definitive references for
viii Preface
doctoral students and experienced researchers in this field, such as those by Parr and
Yang (1989) and Martin (2004). Textbooks in this second category tend to address
a relatively specialized audience, and are slightly too advanced for undergraduate
students. The present textbook is meant to fill the gap between these two categories,
by presenting density functional theory and first-principles materials modelling in a
waywhichshouldbeaccessibletoundergraduates.Inthissense,theauthor’shopeisto
bridgebetweenelementarynotionsofquantummechanics,thetheoryofmaterialsand
first-principles calculations, starting from a minimal set of prerequisites and without
requiring a formal training in advanced theoretical physics.
The presentation in this book revolves around the idea that it is possible to formulate
atheoryofmaterialswherebyapparentlyunrelatedpropertiescanberationalizedand
quantified within a single mathematical model, namely the Schro¨dinger equation. For
thisreasonemphasisisplacedontheunifyingconcepts,andsimpleheuristicarguments
are provided whenever possible. Formal derivations are also given in many cases, in
order to convince the reader that there is no need to formulate a new theory for each
material property, and that in many cases we can find answers in the common root
represented by the Schro¨dinger equation.
The inspiring principle of this book is borrowed from one of the slogans of the Perl
programminglanguage,‘Easythingsshouldbeeasyandhardthingsshouldbepossible’
(L.Wall).Inthepresentcontext,wheneverapropertycanbeunderstoodusingsimple
intuitiveargumentswewillmakenoattemptsatarigorousmathematicaljustification.
Conversely, whenever it seems useful to dissect the mathematical formalism, we will
go through the derivations in order to convince ourselves that there is nothing to be
afraid of.
The presentation style is somewhat cross-disciplinary, insofar as an attempt is made
to seamlessly combine materials science, quantum mechanics, electrodynamics and
numerical analysis, without using a compartmentalized approach. This choice truly
reflects the spirit of first-principles materials modelling, which finds its place at
the intersection between traditional disciplines such as materials science, physics,
chemistry and modern high-performance computing and software engineering.
Throughout the book emphasis is placed on numerical values. One of the greatest
achievements of the theory addressed in this book is to make it possible, in many
cases, to accurately predict the outcomes of actual measurements. This feature is
perhaps the most distinctive and unique strength of density functional theory, as it is
precisely the evolutionary step from qualitative theories of materials to quantitative
predictions of their properties that marked the beginning of first-principles materials
modelling.Thisbooktriestoreflecttheimportancethatweattachtonumericalvalues
by relentlessly calculating, comparing and discussing numbers. Numbers allow us to
develop meaningful approximations to the complex equations of quantum mechanics,
and numbers are at the origin of more abstract conceptual developments. A theory of
materialsisinthefirstplaceatheoryaboutquantitiesthatcanbemeasured;henceit
isimportanttodevelopasenseofthenumericalvaluescorrespondingtotheproperties
discussed in each chapter.
Preface ix
Aneffortwasmadethroughoutthebooktoprovidereferencestotheoriginalliterature
whenever possible, as opposed to referring the reader to other books. This choice was
motivated by the fact that, owing to the digitization and online archiving of many
scientific journals, we now have almost complete and direct access to original papers
datingbacktotheworksofBohronthestructureoftheatomin1913.Allofthejournal
referencescited(over350)areaccessiblewithinastandarduniversitysubscription.All
thecitationstotheoriginalsourcesarehyper-referenced,insuchawaythatthereaders
of the e-book version will be just one click away from the sources.
The choice of referring to the original works was meant to show how the topics
discussed in this book do not form a static piece of knowledge, but are made up
of a myriad of contributions spanning over a century of science. It was felt that even
only glancing at some of the references provided in this book, taking a look at the
language,theequationsandthepresentationstyle,wouldbehelpfulforunderstanding
how certain aspects of the theory of materials were developed, and how complex and
non-linearisthewaytoachieveaformaltheorywhichisbotheffectiveandaesthetically
satisfactory.
An attempt was also made to avoid as much as possible sweeping the more
difficult concepts ‘under the carpet’. While this is somewhat standard practice in
undergraduate education owing to obvious time constraints, it usually generates
significant confusion, and may give the wrong impression that the theory of materials
is a set of rules lacking a unifying principle. In those cases where certain notions
requiretooadvancedadiscussiontofitinthisbook,referencestomorecomprehensive
presentations are always provided.
Similarly to the references, also all the equations and sections are hyper-referenced.
This should make it easier for the e-book readers to hop back and forth between
equations while following some of the derivations.
Exercises are provided within each chapter, and serve the dual purpose of making
the study of the subject more interactive, and exploring some more advanced aspects
whicharenotexplicitlyaddressedinthemaintext.Mostexercisesaredesignedinsuch
a way as to guide the reader through each step, and important intermediate and final
expressions are always provided for cross-checking. The exercises are scattered within
the chapters, instead of being collected at the end as in most textbooks. This should
motivate the reader to actually put some effort into solving the exercises, without
being discouraged by the more traditional lists of questions at the end of chapters.
The assignments in each exercise are clearly identified by a symbol (◮) in order to
distinguish them from those parts which form complements to the theory in the main
text ((cid:3)). Some exercises are intended to give a flavour of the key steps involved in
actualfirst-principlescalculationsofmaterialproperties.Someothersareincludedfor
acquiring familiarity with techniques of calculus and numerical methods which may
not have been introduced in previous undergraduate courses. During a first reading of
this book it may be advantageous to skip the exercises altogether, and then go back
and try them during a second reading.
It goes without saying that some of the topics discussed in this book could be
presented more rigorously and elegantly using advanced conceptual tools, such as
perturbation theory and second quantization. However, such an approach would defy