Linköping Studies in Science and Technology Dissertation No. 1453
Thermal properties of materials from first principles
Olle Hellman
Linköping University
Department of Physics, Chemistry and Biology Theory and Modelling
SE-581 83 Linköping, Sweden Linköping 2012
Typeset using LTEX
Abstract
In the search of clean and efficient energy sources intermediate tempera-ture solid oxide fuel cells are among the prime candidates. What sets the limit of their efficiency is the solid electrolyte. A promising material for the electrolyte is ceria. This thesis aims to improve the characteristics of these electrolytes and help provide thorough physical understanding of the processes involved. This is realised using first principles calculations. The class of methods based on density functional theory generally ig-nores temperature effects. To accurately describe the intermediate tem-perature characteristics I have made adjustments to existing frameworks and developed a qualitatively new method. The new technique, the high temperature effective potential method, is a general theory. The validity is proven on a number of model systems.
Other subprojects include low-dimensional segregation effects, ad-justments to defect concentration formalism and optimisations of ionic conductivity.
Det finns stort intresse i att hitta rena och effektiva energikällor. Fasta ox-idbränsleceller är ett möjligt sätt att åstadkomma detta. I dessa konvert-eras kemisk energi till elektricitet och värme i en kontrollerad reaktion. Det uppstår inga av de giftiga biprodukter som alltid förekommer i av-gaser från förbränningsprocesser. Bränslecellerna är också effektiva: upp till 60% effektivitet vid framställning av el och över 80% om man även tar tillvara på spillvärmen. För att göra dessa till kommersiellt gångbara produkter behöver de dock bli effektivare och billigare.
Flaskhalsen i bränslecellerna är elektrolytmaterialet. Elektrolyten är elektriskt isolerande och leder syrejoner från luften till bränslet. Jag har främst undersökt ceriumdioxid, vilket är en lovande kandidat för dessa material. Metoden som använts är kvantmekaniska beräkningar inom materialteori. Det är en bred klass av metoder, alla baserade på täthets-funktionalteori. Det är komplicerade beräkningar där man av nöd är tvungen att införa approximationer. En vanlig sådan är att ignorera tem-peratureffekter. För material avsedda att användas i rumstemperatur har det minimal effekt. De bränsleceller vi pratar om opererar mellan 600 och 1000°C. I detta temperaturspann kan man inte längre ignorera effekten av värme.
För att bättre beskriva material vid höga temperaturer har jag utveck-lat en ny metodik. Det är en allmän metod för att bestämma termody-namiska potentialer, som jag bekräftar med flera exempel. Andra pro-jekt har ämnats åt segregationseffekter i lågdimensionella system, för-bättringar av den formalism som används för att beskriva defekter och optimering av jonledningförmåga i ceriumdioxid.
Preface
This work was carried out in the theoretical physics group at Linköping university between 2007 and 2012. There are a number of people whom I could not have done this without. First and foremost I would like to thank my supervisor, Sergei Simak, for imparting knowledge and endur-ing support durendur-ing my forays further and further away from the initial goals, and my secondary supervisor Igor Abrikosov for his attempts at reining me in. I am grateful to Natalia Skorodumova for present and fu-ture collaborations and advice. I would like to thank my colleagues and friends in the Theoretical Physics and Computational physics groups for making the time spent at work seem nothing like a chore. I also want to thank my family for support, especially my sister for tirelessly proofread-ing incomprehensible manuscripts and min ömma moder for cookies. Olle Hellman
C O N T E N T S
Notation xi Acronyms xii 1 Introduction 3
1.1 Outline of thesis . . . 4
2 First principles calculations 5 2.1 Density functional theory . . . 5
2.2 Exchange-correlation . . . 8
2.3 Pseudopotentials and projector augmented waves . . . 9
2.4 Strongly correlated systems . . . 10
3 Lattice dynamics 15 3.1 The harmonic approximation . . . 15
3.2 Dealing with anharmonism . . . 19
4 Molecular dynamics 21 4.1 Born-Oppenheimer molecular dynamics . . . 21
4.2 Ergodicity and phase space sampling . . . 22
4.3 Ensembles and thermostats . . . 25
4.4 Free energy and thermodynamic integration . . . 26
5 Temperature dependent effective potential 29 5.1 A one-dimensional anharmonic oscillator . . . 29
5.2 Extending the method to the periodic lattice . . . 32
5.3 Symmetry constrained force constant extraction . . . 34
5.4 Internal degrees of freedom . . . 37
5.5 Determining the free energy . . . 38
5.6 Upsampling and error analysis . . . 41
5.7 Computational efficiency . . . 43
5.8 Practical procedure . . . 45
6 Monte Carlo method 49 6.1 The Metropolis algorithm . . . 49
6.2 Model hamiltonian . . . 50
7 Thermal defects 51 7.1 Energetics of defect formation . . . 51
7.2 Oxygen vacancies in ceria . . . 51
7.3 Vacancy migration . . . 53 8 Conclusions 55
Bibliography 57 List of figures 62
Paper I 65
Lattice dynamics of anharmonic solids from first principles Paper II 71
Double-segregation effect in AgxPd1−x/Ru(0001) thin film nanostructures
Paper III 81
Charge Redistribution Mechanisms of Ceria Reduction Paper IV 87
Tuning ionic conductivity in ceria by volume optimization Paper V 93
Temperature dependent effective potential method for accurate free energy calculation of solids
Paper VI 103
Temperature dependence of elastic constants of TiN from ab initio molecular dynamics simulations
N O T A T I O N
notation a( ) function
a[ ] functional {a} set of all a
¯ ¯ a matrix ˆ a operator a vector constants Z atomic number
~ reduced Planck constant e electron charge
kB Boltzmann constant
coordinates
r Cartesian coordinate R lattice vector
τ equilibrium position in the unit cell u displacement from equilibrium
position p momentum v velocity f force energies EU LDA+U energy EH Hartree energy T kinetic energy Et total energy Exc exchange-correlation energy masses M mass of atom me electron rest mass
operators ˆ
H Hamiltonian operator ˆH = ˆT + V ˆ
T kinetic energy operator phonon formalism
ω angular frequency q wave vector
Φ force constant matrix potentials
VU LDA+U Coloumb potential Veff effective potential
Vext exchange-correlation potential VH Hartree potential
Vxc exchange-correlation potential thermodynamics
F Helmholtz free energy G Gibbs free energy
S entropy T temperature U internal energy V volume
Λ thermal De Broglie wavelength β thermodynamic beta Z partition function p pressure wavefunctions Ψ many-body wavefunction φ single-particle wavefunction ψ atomic-like orbitals n electron density
bcc body centered cubic
BOMD Born-Oppenheimer molecular dynamics DFT density functional theory
DMFT dynamical mean-field theory fcc face centered cubic
GGA generalized gradient approximation hcp hexagonal close packed
LDA local density approximation MD molecular dynamics PAW projector augmented wave SOFC solid oxide fuel cells
The first rule of thermodynamics is you do not talk about thermodynamics.
FIGURE 1.1: Schematic for the operation of a solid oxide fuel cell. Oxygen gas is split at the cathode and travels as oxygen ions through the electrolyte towards the anode. There it is combined with hydrogen. The electrolyte is electrically insu-lating and energy is released as a current.37
I N T R O D U C T I O N
There is a global demand for clean and efficient energy sources. One such source could be intermediate temperature solid oxide fuel cells (SOFC).58These devices produce electricity by directly
oxidising fuel with no toxic by-products. The efficiency can be over 60% when generating electricity, and over 80% when the waste heat is taken care of.15In figure 1.1 the operation of a SOFC is illustrated. The solid electrolyte that provides the conduction of oxygen ions is the primary bottleneck, both in terms of effi-ciency and cost. A substantial portion of the work in this thesis is devoted to the improvement and understanding of ceria-based electrolytes.
We have based our work on ab initio calculations. It is a class of theoretical methods where the input is the fundamental laws of physics and no assumptions or empirical models are made. When practically using these techniques, some approximations have to be introduced. Temperature effects, for example, are usually ignored or crudely treated. This is often an appropriate approximation, the operational temperatures for most materials are around room temperature where the finite temperature ef-fects are small. However, the operational temperatures of SOFC
e -Anode Cathode Electrolyte Electric Current Air In Fuel In e -e -e -Excess Fuel and
Water Gases OutUnused
H2O
O
2-O2
2-are 800–1000°C. In this range the temperature effects cannot be ignored.
During the project we realised that the theoretical framework for dealing with temperature effects, in particular thermodynamic stabilities, was not up to par. There was a host of pre computer-era work extensively detailing how to theoretically treat temper-ature effects. We sidetracked a bit from the initial task and devel-oped a thorough formalism to accurately determine finite tem-perature properties using state-of-the-art methods, bridging the gap between formal theory and practical implementation. This has lead to a qualitatively new way of determining relevant ther-modynamic potentials.
Other than temperature, we have faced other issues when it comes to first principles calculations. The scales are limited. In size, we can simulate features up to a few Ångström and in time we talk about picoseconds at most. To extend our reach we ex-tract parameters from first principles calculations and build mod-els on a relevant scales. We have used this approach to study low-dimensional segregation effects as well as seeking to explain the general behaviour of the ionic conductivity in ceria.
Outline of thesis
In chapter 2 I outline the basics of first principles calculations. Chapter 3 deals with lattice dynamics. I then proceed to dis-cuss molecular dynamics in chapter 4. In chapter 5 I describe the novel temperature dependent effective potential method. The fi-nal two chapters, 6 and 7 deal with the Monte-Carlo method and crystal defects respectively. The papers and the authors contri-butions are presented at the end.
(I)In this thesis I do not discuss the issues of relativistic quan-tum mechanics and accordingly do not adress the relativistic Dirac equation.18
F I R S T P R I N C I P L E S C A L C U L A T I O N S
At the turn of the 20th century, the development of a theoretical framework describing matter on the microscopic scale was in full swing. In 1926, Erwin Schrödinger formulated his famous equa-tion,55which if solved allows us to describe any physical system at the atomic scale.I It opened up the possibility to model a
sys-tem without any empirical parameters. This class of calculations are called first principles calculations.
Density functional theory
Ideally, to investigate a system of N electrons and M nuclei, we begin with the Schrödinger equation:
ˆ
HΨ = EΨ (2.1)
where
Ψ = Ψ(r1, . . . , rN, rn1, . . . , rnM) (2.2)
is a wavefunction that depends on the position of all electrons {r} and nuclei {rn
}. The Hamiltonian takes this form: ˆ H =− ~ 2 2me X i ∇2i−~ 2 2 X k ∇2 k Mk +1 2 X i6=j e2 |ri− rj| +1 2 X k6=l ZkZl |rn k− rnl|− X i,k eZk |ri− rnk| . (2.3)
The resulting equations can only be exactly solved for the sim-plest of systems. To solve it we successively apply layers of ap-proximations, hopefully maintaining physical relevance. The first step is to separate the nuclei and the electronic degrees of free-dom. This is the Born-Oppenheimer approximation.12The nuclei
terms will enter as a constant. We are left with ˆ H =− ~ 2 2me X i ∇2i | {z } ˆ T +1 2 X i6=j e2 |ri− rj| | {z } VH −X i,k eZk |ri− rnk| | {z } Vext . (2.4)
This is still too complex to solve. In 1964 Hohenberg and Kohn31 formulated the density functional theory (DFT). The central idea is to replace the many-body problem in equation 2.4 with an equation for the electron density. The density, with its depen-dence on three spatial coordinates, will implicitly contain the electronic degrees of freedom. Their original formulation con-tains two theorems stating this is possible:
(II)The unknown parts are the kinetic and exchange-correlation energy functionals. There were some early at-tempts at approximating the kinetic energy functional, origi-nating from Thomas-Fermi the-ory.60,22It approximated the
kinetic energy to that of a uni-form electron gas of the same density. For that specific case the theory is exact, but it fails to predict bonding and thus the existence of molecules. It was later corrected by von Weiszäcker63but this
formula-tion was overbinding. Orbital free density functional theory has been an active field of research ever since, with nu-merous functionals suggested. Unfortunately none of these have been as robust as the Kohn-Sham theory and have limited practical applications.
1. The potential Vextis determined uniqely by the ground state
density n, up to an arbitrary constant.
2. Any external potential has a universal total energy func-tional FHK. The exact ground state density minimizes the
total energy functional Et[n].
The Hohenberg-Kohn functional is defined as FHK[n] = Et[n]−
Z
Vext[n]ndr, (2.5)
where Et[n] is the total energy functional. This functional
ex-ists, but the explicit form is unknown.II The Kohn-Sham equa-tions,41formulated in 1965, provide the framework wherein the electronic structure problem in the density functional formula-tion becomes solvable. Here we follow the reasoning in Eshrig.20 The Kohn-Sham ansatz is to replace the real electrons with non-interacting quasiparticles that produce the correct ground state density, and gather all the unknown terms in one term called exchange and correlation. The universal functional now takes the form
FHK[n] = T [n] + EH[n] + Exc[n]−
Z
Vext[n]ndr. (2.6)
It is divided into kinetic energy (T ), Hartree energy (EH), the
ex-ternal potential (Vext) and a term containing all the many-body
terms, Exc. The exact form of the exchange-correlation is not
known, but later we will see that it can be approximated. We can now define the total energy functional
Et[n] = T [n] + EH[n] + Exc[n] +
Z
Vext(r)n(r)dr. (2.7)
We want to find the density that minimises this energy func-tional. In order to do so we perform a constrained search for the stationary points, with the constant particle number N as the constraint and Lagrange multipliers i:
L =Et[n]− i Z n(r)dr− N , (2.8) δL = Z δ L δnδndr = 0. (2.9)
At this point we take advantage of the previously mentioned Kohn-Sham ansatz. We denote the ground state many-body wave-function for the non-interacting quasiparticles Ψ0. This can al-ways be written as a determinantal state of one-particle wave-functions:
Ψ0=√1
D E N S I T Y F U N C T I O N A L T H E O R Y
and the density expressed in these orbitals:
n =X i φ∗iφi δn =X i δφ∗iφi+ φ∗iδφ (2.11)
and thus define the kinetic energy functional as T [n] =1
2 X
i
hφi| ˆT|φii . (2.12)
Performing a variation on T with respect to φ∗jgives me
δT = T [n + δφ∗j]− T [n] = =φj+ δφ∗j ˆT|φji − hφj| ˆT|φji = = Z ˆ Tφjδφ∗jdr = Z δT δφ∗ j δφ∗jdr, which tells me δT δφ∗ i = ˆTφi= Z δT δn(r0) δn(r0) δφ∗ i(r) dr0=δT δnφi. (2.13) Putting equation 2.11 into 2.9 gives
δL =X i Z δ L δnφi | {z } =0 δφ∗idr + X i Z δ L δnφ ∗ i | {z } =0 δφdr = 0. (2.14)
This will give us constraints on the orbitals: δL δnφi= δT δn+ δEH δn + δExc δn + Vext− i φi= 0. (2.15)
Looking at these terms we find the Hartree potential δEH[n] δn = Z n(r0) |r0− r|dr 0= V H[n] (2.16) and δExc[n] δn = Vxc[n] (2.17)
as the definition of the exchange-correlation potential. Combin-ing these terms we have
ˆT + VH+ Vxc+ Vext− i
φi= 0. (2.18)
By putting the potentials into one term, Veff= VH+ Vxc+ Vext, we
have the Schrödinger-like Kohn-Sham equations for the quasi-particle orbitals φithat produce the ground state density
ˆT + Veff[n]
One can then show that the total energy functional becomes Et[n] = X i i+ Z n(r)Veff[n]dr. (2.20)
We can now solve equations 2.11, 2.19 and 2.20 self-consistently and find the electronic ground state density.
We have not yet mentioned spin-polarised densities. If we knew the exact Excand solved the Kohn-Sham equations an
oxy-gen atom, for example, density functional theory would produce the correct ground state energy and density, but it could not tell us anything about the spin polarisation. In practise, when using approximate Exc, one has to explicitly introduce spin polarised
densities and spin dependent Excto get accurate electronic
struc-ture for magnetic systems. The Kohn-Sham equations for this case can be obtained in a similar manner.
Exchange-correlation
So far we have applied no approximations, the theory is still ex-akt. To practically solve the equations the unknown Vxchas to be
defined somehow. The local density approximation (LDA) was introduced at the same time as the Kohn-Sham equations.41 It
takes the form
ELDA xc =
Z
(n(r))n(r)dr, (2.21)
where is the exchange-correlation of a uniform electron gas, something that can be calculated with the quantum Monte Carlo method. This is a very successful approximation, yielding good results for a broad range of materials. The success can be at-tributed to the reasonable treatment of the exchange-correlation hole and cancellation of errors.14The natural extension to this is to include the gradient of the density, and obtain the exchange and correlation from a uniform electron gas with a gradient. This is the generalized gradient approximation (GGA),51 which has
the form
EGGA xc =
Z
f (n(r),∇n(r))n(r)dr. (2.22) There is no unique way to determine the gradient dependence of the uniform electron gas, that is why there are several forms of GGA optimized for different cases.
There are notable limitations to these functionals. For exam-ple: the band gap of semiconductors is underestimated (and some-times outright missing, predicting a metallic state). In addition the LDA underestimates lattice parameters, whereas the GGA overestimates them.
P S E U D O P O T E N T I A L S A N D P R O J E C T O R A U G M E N T E D W AV E S
Pseudopotentials and projector augmented waves
When solving the Kohn-Sham equations for a crystalline solid the resulting wavefunctions oscillate rapidly around the nuclei and vary more slowly in the interstitial regions. To solve the equations numerically one needs to use a basis set. It is a non-trivial task to find basis functions that can describe both the re-gion around the nuclei, where atomic orbitals would work well, as well as the interstitial, where plane waves are more suitable. It would also be beneficial if one could find a way to ignore the deep core states. These states do not contribute much to bond-ing and are weakly affected by it. The computational complexity usually scales asO(N3), with N being the number of electrons.
Any electron that can be omitted would greatly increase the com-putational efficiency.
The pseudopotential approach was introduced by Hellmann in 193530but was put in the modern form by Phillips and Klein-man in 1959.53They showed that one can construct a modified
valence wave function ˜φv using the true core φc and valence
wave functions φv ˜ φv E =|φvi + X c acv|φci , acv= D φc ˜ φv E . (2.23)
This pseudo-wavefunction satisfy ˆ H +X c (v− c)|φci hφc| ! φ˜v E = v φ˜v E . (2.24)
That is the pseudowavefunctions have the same eigenvalues as the true wavefunctions. What is added is an extra potential term in the Hamiltonian. This early attempt showed that the division between core and valence states is possible, and if choosing fixed core states the number of electrons in the problem at hand will be dramatically reduced. There are many pseudopotential ap-proaches, suited for different tasks.45,40
We have used the projector augmented wave (PAW)9method throughout this thesis. This is a modern approach that keeps the all-electron wave function via a linear transformation
T = 1 +X
c
Tc, (2.25)
whereTconly acts within a region around nuclei c. This
trans-formation can be applied to any operator ˜
A =T−1AˆT (2.26)
allowing it to work on the smooth pseudowaves but keeping the information from the full wavefunction. This increases the trans-ferability of the potentials.
distance from nucleus (r/a0) 0 1 2 3 U=‒5.0 U=‒2.5 U=0.0 U=2.5 U=5.0 FIGURE 2.1: An illustration of the effect of LDA+U. Presented areρ(r)r2versusr/a
0for a
hydrogen atom from LDA+U-calculations (dashed lines) as compared to the analytical so-lution (solid grey lines). Pure LDA corresponds toU = 0. The near perfect agreement of the Bohr radius at negative Uis because LDA overesti-mates correlation for a single electron, it should be 0. Al-though the Bohr radius can be corrected by LDA+U, the asymptotic behaviour cannot.
Strongly correlated systems
In strongly correlated systems, where the inherent many-body ef-fects are crucial, the local density-based functionals fails to model the systems completely. The effects cannot be described well with non-interacting quasiparticles. These materials include, but are not limited to, transition metal oxides, Mott insulators and cuprates. Hubbard35,36 introduced the first attempts at solving these problems. Presently, the methods most widely used are LDA+U,5 hybrid functionals52,7and dynamical mean-field the-ory (DMFT).4 In this thesis we have exclusively used LDA+U. Although it is a simple model, it is well suited for the problems in question. The main strength is the minimal added computa-tional complexity, making it usable for large systems. The major drawback is the need for adjustable parameters as well as the fact that it is an inherently one-electron theory and can never describe strongly correlated systems where electrons simultane-ously show localised and itinerant behaviour. DMFT or hybrid functionals would solve this, but for our systems of interest the added computational complexity unfortunately renders it prac-tically impossible to use them. The principle of LDA+U is rather simple. We will motivate it in the rotationally invariant case via the variational principle, as detailed in Eshrig.21Add an extra
term to equation 2.7:
Et[n] = T [n]+EH[n]+Exc[n]+EU[φ, o]+
Z
Vext(r)n(r)dr (2.27)
that depends on the Kohn-Sham orbitals and their occupation numbers o. When varying this expression with respect to φ∗i to
get the Kohn-Sham equations, an extra term will appear: δEU δφ∗ i =X jµσ ∂EU ∂ ˜oµσ δ˜oµσ δφ∗ i (2.28) where ∂EU ∂ ˜oµσ =VU(µ, σ) (2.29) δ˜oµσ δφ∗ i =|ψi hψ|φi . (2.30)
j is a site, µ is an orbital quantum number and σ is a spin quan-tum number. EUwill depend a projection onto correlated local
orbitals|jµσi = |ψi. This projection is given by local orbital oc-cupation numbers, ˜oµσ, that depend on the Kohn-Sham orbitals
and occupation numbers. We choose to describe the angular component of the strongly correlated electrons with eigenstates to angular momentum, centered at site i:
|ψii = |miσii
mi=−li, . . . , li
σi=↑, ↓ .
S T R O N G L Y C O R R E L A T E D S Y S T E M S
We only consider interaction within the site. That gives the form of the expectation value for the screened electron-electron inter-action:
hm1m2| Vee|m3m4i
σ1= σ3, σ2= σ4, Vee≈ Vee(|r1− r2|).
(2.32) The dependence on solely|r1− r2| is of course an
approxima-tion. We investigate how these matrix elements transform under a rotation. R( ˆO) is the unitary transformation for the rotation ˆO. hm1m2| Vee|m3m4i = X m0 1m02m03m04 Rm† 1m0 1( ˆO)R † m2m0 2( ˆO)× m0 1m02 Vee m03m04 Rm3m03( ˆO)Rm4m04( ˆO) and Z Rm1m2( ˆO)R†m3m4( ˆO)d ˆO = 1 2l + 1δm1m4δm2m4 (2.33) Z d ˆO =1. (2.34)
Using these relations we integrate over all rotations and look at: X m1 hm1m2| Vee|m1m4i =X m1 Z Z hm1m2| Vee|m1m4i d ˆOd ˆO0 =X m1 Z Z X m0 1m02m03m04 R†m 1m01( ˆO 0)R† m2m02( ˆO) m0 1m02 Vee m03m04 Rm0 3m1( ˆO 0)R m0 4m4( ˆO)d ˆOd ˆO 0 = Z X m0 2m04 Z X m1m01m03 Rm† 1m01( ˆO 0)Rm0 3m1( ˆO0)d ˆO 0 | {z } δm1m1δ m03m0 1 R†m 2m02( ˆO) m0 1m02 Vee m03m04 Rm0 4m4( ˆO)d ˆO = Z X m0 1m02m04 R†m 2m02( ˆO)m 0 1m02 Vee m01m04 Rm0 4m4( ˆO)d ˆO = 1 2l + 1 X m0 1m02m04 m0 1m02 Vee m01m04 δm2m4δm0 2m04 = 1 2l + 1δm2m4 X m0 1m02 m0 1m02 Vee m01m02 = (2l + 1)U δm2m4.
Nd f-states O p-states FIGURE 2.2: Orbital projected density of states for Nd2O3
for different values of U. The plots are filled up to the high-est occupied state. The striped portions indicate empty states. With noUwe have a metal-lic solution which is clearly unphysical.U = 2gives a reasonable electronic struc-ture whereas a too high value pushes the occupied f-bands into the valence band.
DO S Energy (eV) -2 -1 0 1 2 3 4 -3 U= 0 U=2 U=5
In the same manner one gets
m1
m1m2| Vee|m3m1 = δm2m3(U + 2lJ). (2.35)
These sum rules gives an expression for the energy that depends only on U and J. The following steps depend on the specific implementation. As it stands it follows Dudarev:19First one has to build the density matrix on each site i:
ρσij= oσioσj
oσi =
nσ(r)ψi(r)dr
where oiis the occupation of states projected onto the density at
site i using the atomic orbitals ψi. The energy functional is then
ELDA+U= ELDA+ U− J 2 σ i ρσii− ij ρσijρσij (2.36)
and the corresponding U -potential is
VUij= (U− J) 1 2δij− ρ σ ij . (2.37)
Since only the difference between U and J enters the calculations one usually denotes U−J just U. Although the method is named
LDA+U it can equally well be used together with GGA.
Finally, it only remains to determine the value of U . Although it can in principle be calculated from restricted LDA calculations6 it is usually fitted to some experimental value.3The extra
poten-tial will shift the Kohn-Sham eigenvalues according to
LDA+Ui = LDAi + U
1 2− oi
S T R O N G L Y C O R R E L A T E D S Y S T E M S
that is occupied states will have lower energy and unoccupied will be shifted up. For example, rare earth oxides with incom-pletely filled 4f -shells are problematic in the LDA. By applying U one can obtain the correct semiconductor ground state. Figure 2.2 shows the density of states of Nd2O3for different values of
U .
The ideas presented in sections 2.1–2.4 are by no means a com-plete description of the field. They illustrate how the generality and simplicity of expressions such as equation 2.1 has to make way for long and complex formulations when it comes to actual implementation. They are classes approximations: exchange-correlation being one out of necessity since the exact form is un-known, in contrast to pseudopotentials that are introduced for practical reasons to decrease the computational complexity. The LDA+U formalism is a good example of how to increase the long-evity of the density functional theory. While it is a crude and nearly phenomenological approach, it extends the reach of DFT to systems it was never imagined for.
L A T T I C E D Y N A M I C S
What most first principles ultimately come down to is the com-parison of energies, be it configurational energies, surface en-ergies, mixing enthalpies or lattice stabilities. What is usually compared are total energies. That is appropriate when the effects of temperature can be neglected, if the temperate effects are not negligible it is the Gibbs free energy that should be compared if the temperature T and pressure p are external conditions:
G = U− T S + pV . (3.1)
Modern DFT technique give us a very efficient way of determin-ing U , the internal energy and Se, the electronic contribution to
the entropy via the Mermin functional.47The thermal properties of the lattice, the major part of S, are more complicated.
At the beginning of the 20thcentury Born and von Kármán13 introduced the dynamical theory of lattices. It revolves around determining the response of the system to a displaced atom and the resulting vibrations, in essence what happens when you heat the lattice.34,11
The harmonic approximation
The starting point of lattice dynamics is a Taylor expansion in the atomic displacements of the potential energy.34It is convenient to
define the atomic positions as displacements u from their equi-librium positions Ri+ τi:
ri= Ri+ τi+ ui. (3.2)
Riis a lattice vector and τithe position in the unit cell. The
po-tential energy per unit cell of the crystal can then be expanded as U ({u}) =U0+ X i X µ Φµiuµi+ 1 2! X ij X µν Φµνijuµiuνj+ 1 3! X ijk X µνξ Φµνξijkuµiuνjuξk+ . . . (3.3)
0 1 2 3 4 5 6 7 8 9 10 Γ R M Γ X
FIGURE 3.1: Phonon dispersion relationsω(q)for B1 FeV. The freqency is in units of THZ.
where ijk are indices to atoms and µνξ Cartesian indices. The constants Φµi = ∂U ∂uµi u=0 = 0 (3.4) Φµνij = ∂2U ∂uµi∂uν j u=0 (3.5) Φµνξijk = ∂ 3U ∂uµi∂uν j∂u ξ k u=0 (3.6) are called the force constant matrices. The constant term is usu-ally set to the energy of the static lattice. The first order terms have to be zero since we stated that the derivatives are taken at equilibrium positions. That leaves only second and higher or-der terms. If we truncate at the second or-derivatives we have the so-called harmonic approximation. It has the Hamiltonian
ˆ H =X i p2 i 2Mi +X ij uiΦ¯¯ijuj, (3.7)
noting that the second term above involves a sum over pairs. We can rearrange this sum. From each atom i at position ταin the
unit cell we will have a star of vectors to every atom j at position τβ. This set of l vectors we denote{Rαβl } and only depends on
the indices to the positions in the unit cell, αβ, and not on the global atomic indices ij. The Hamiltonian can the be written
ˆ H =X iα p2iα 2Miα +X β X l uiαΦ¯¯αβ(Rαβl )ulβ . (3.8)
Given a sufficiently large crystal,43 large enough that the sur-face effects can be ignored, this expression is independent of the atomic index i. That gives us an equation of motion for each atom in the unit cell:
¨ uαMα=− X β X l ¯ ¯ Φαβ(Rαβl )ulβ. (3.9)
This is a drastic reduction in the number of coupled equations that have to be solved. It is possible to find a coordinate trans-formation that simultaneously diagonalises the momentum and positions since the Hamiltonian is a sum of two positive definite quadratic forms. This canonical transformation is an expansion of u and p in plane waves
uµα= X qj s ~ 2N Mαωj(q) αµj (q)eiq·ταa qj+ a†−qj (3.10) pµα= X q,j 1 i r ~Mαωj(q) 2N αµ j (q)e iq·ταa qj− a†−qj (3.11)
T H E H A R M O N I C A P P R O X I M A T I O N
and their inverse aqj= X lαµ αµj (q) √ 2N~e −iq·τlα uµ αpMαωj(q) + i p µ α pMαωj(q) ! (3.12) a†qj=X lαµ αµj (q) √ 2N~e −iq·τlα uµ αpMαωj(q)− i pµ α pMαωj(q) ! . (3.13) N is the number of atoms in the unit cell. q is a wave vector in the first Brillouin zone. ω2and are the eigenvalues and eigen-vectors of the dynamical matrix
ω2(q)(q) = ¯Φ(q)(q)¯ (3.14) ¯ ¯ Φ(q) = ¯ ¯ Φ11(q) · · · Φ¯¯N 1(q) .. . . .. ... ¯ ¯ ΦN 1(q) · · · Φ¯¯N N(q) (3.15) with ¯ ¯ Φαβ(q) = X l eiq·Rl pMαMβ ¯ ¯ Φαβ(Rl). (3.16)
The vibrational frequency ω is a multivalued function of q. It will have 3N values for each wavevector (some of them can be degen-erate). These are the dispersion relations. Using the orthonor-mality relations of u and p, and switching to quantum operators one realises that we have defined annihilation and creation oper-ators for a quantum mechanical harmonic oscillator. The bosonic quasiparticles that are created are called phonons. The Hamilto-nian takes the remarkably simple form
ˆ H =X qj ~ωj(q) ˆ a†qjˆaqj+ 1 2 , (3.17)
a sum of uncoupled harmonic oscillators. Each have the partition function Zj= ∞ X n=0 e−β(n+12)~ωj = e −β~ωj2 1− e−β~ωj (3.18) that yields the total partition function
Z =Y
jq
e−β~ωjq2
FIGURE 3.2: Illustration of how to go from Helmholtz to Gibbs free energy. Helmholtz free energy in the right panel is determined on a regular grid of volumes and temperatures. The pressure in the left panel is given by p = −∂F ∂V T .
The pressure at each point yields a one-to-one map-ping between the free ener-gies. Note how the pressure-temperature grid is no longer evenly spaced. This particular example is for Al in the face centered cubic (fcc) phase.
13 14 15 16 17 18 19 20 100 200 300 400 500 600 700 800 900 1000 -10 -5 0 5 10 15 20 25 30 100 200 300 400 500 600 700 800 900 1000
Pressure (GPa) Volume (Å3)
Te rm pe ra tu re ( K) Te rm pe ra tu re ( K)
Gibbs free energy Helmholtz free energy
From this we can get, for instance, the Helmholtz free energy
F =− kBT lnZ = jq ωjq 2 + kBT ln 1− exp − ωk jq BT = ∞ 0 g(ω) ω 2 + g(ω)kBT ln 1− exp −k ω BT dω. (3.20) The transition from a sum over wave vectors and branches to an integral over the phonon density of states is out of convenience. The phonon density of states is given by
g(ω) = j BZ δ (ω− ωj(q)) 2π3 dq. (3.21)
Explicit knowledge of the partition function is very useful. In ad-dition to the Helmholtz free energy we can, among other things, determine the vibrational entropy and constant volume heat ca-pacity.
There are several limitations in the harmonic approximation. Thermal expansion, for example, is absent. This can be remedied by using the harmonic approximation for several volumes and temperatures. This is called the quasi-harmonic approximation, and can provide a mapping from the Helmholtz to the Gibbs free energy, as illustrated in figure 3.2.
The omission of anharmonic terms can have serious conse-quences. If any of the second derivatives in equation 3.5 are negative they will carry through to the dynamical matrix and
D E A L I N G W I T H A N H A R M O N I S M
the eigenvalues ω2will become negative. This indicates that the crystal is dynamically unstable. In reality the crystal can be sta-bilised by the anharmonic contribution. To compare the free en-ergies of two competing crystal structures is impossible if one of them is dynamically unstable at zero temperature, since the free energy is not defined for imaginary frequencies.
Dealing with anharmonism
Traditionally the problem of dynamical instability was addressed either by a self-consistent approach or including more terms in the expansion.39,10,11Born and Hooton made the realisation that even though the second derivatives at the equilibrium position are negative, the atoms rarely occupy these positions. They move in the effective potential of their non-stationary neighbours. By sampling the potential energy surface not at the equilibrium po-sitions but at the probable popo-sitions for a given temperature we get a harmonic approximation that describes the system at ele-vated temperatures.
The self-consistent formalism uses an iterative procedure.26
By creating a harmonic potential that is used to create thermal excitations that give a new harmonic potential. This is repeated until self-consistency.
The double-time Greens functions, developed by Choquard,17
use a cumulant expansion in the higher order terms. Although formally exact, it requires knowledge of the higher order force constants. Obtaining these accurately from from first principles calculations for something other than the simplest of systems is computationally very demanding.16
A recent approach by Souvatzis56uses ab initio supercell cal-culations and a self-consistent formalism. A problem with this approach is that the excitations can only be done in the harmonic sense which means probing phase space with a limited basis set. Where the harmonic approximation works well this is not a prob-lem. For strongly anharmonic systems a harmonic basis set is by definition a bad way of describing the dynamics of the system. A harmonic Hamiltonian will probe the wrong part of phase space. I have developed a new method that is similar in spirit to Born and Hootons32original idea, but with a foundation in ab initio molecular dynamics that overcomes the limitations of ear-lier techniques, described in chapter 5.
M O L E C U L A R D Y N A M I C S
The phonon formalism, detailed in the previous chapter, is well suited to describe averaged thermal quantities. It provides a model how the thermal excitations behave but is limited in de-scribing the actual dynamics of the system. Diffusion, for exam-ple, cannot be described with phonon formalism.
Molecular dynamics (MD) gives explicit knowledge about the movement of atoms at finite temperature. This technique solves the equations of motion for the atoms and produces the trajec-tories over time. The simulation is an idealised version of an experiment with few constraints on the information that can be accessed.
Born-Oppenheimer molecular dynamics
To perform an MD simulation one needs a way to determine in-teratomic forces. Historically this has been done using empiri-cal potentials. These have yielded good results in the past but are severely limited in terms of transferability, and the fact that electrons are completely abstracted. They have the advantage of being computationally very efficient, which is why I have used them throughout this thesis for illustration purposes.
When investigating properties of materials at finite tempera-ture we have exclusively used Born-Oppenheimer molecular dy-namics (BOMD), where the interatomic forces are calculated us-ing quantum mechanics. When discussus-ing DFT in the previous chapters the first approximation was to separate the nuclei and electronic degrees of freedom. In the same way that the nuclei can be considered stationary from the electronic point of view the nuclei can be seen as classical particles moving in a potential created by the electrons. This is provided the relaxation time for the electrons is sufficiently small, which is usually the case. With the Hellmann-Feynmann23theorem
dE dλ =hΨ|
d ˆH
dλ |Ψi (4.1)
one can obtain the forces fi acting on each atom. The classical
equations of motion
˙ri= pi
Mi
˙pi= fi
(4.2)
are 6N coupled differential equations. Since we calculate forces at instantaneous positions these have to be solved iteratively. We
use the velocity Verlet algorithm:62 ri(t + ∆t) = ri(t) + ˙ri(t)∆t +1 2¨ri(t)∆t 2 (4.3) ˙ri(t + ∆t) = ˙ri(t) + ¨ri(t + ∆t) + ¨ri(t) 2 ∆t. (4.4)
Here ∆t is the step discretising time. Given initial positions and velocities, the forces are given by DFT calculations and the sys-tem is propagated from time t to time t + ∆t. After an appropri-ate number of steps, statistics and other properties of interest are gathered.
While this essentially describes the technique there is still a wealth of things to consider. The classical molecular dynam-ics, that govern the motion of our nuclei, are well reviewed in Allen and Tildesley1, Hansen and McDonald28and Frenkel and Smit25among others. The intricacies of ab initio molecular
dy-namics is thorougly described in Marx and Hutter.46In the rest of this chapter will touch upon some key points that are often overlooked in the literature.
Ergodicity and phase space sampling
The apparent straightforwardness of molecular dynamics can be misleading. The formalism for the classical trajectories were orig-inally developed for classical systems of massive sizes (as com-pared to the sizes accessible with ab initio calculations). The finite size effects can, if one is not careful, have unpredictable consequences. The finite size applies not only to the number of atoms but also to the limited simulation time. A limited simula-tion time severely affects the quality of averaged quantities.
The average value of property a in thermal equilibrium is given by the ensemble average:
hai =R a({p}, {q}) exp (−βH({p}, {q})) d{p}d{q}R exp (−βH({p}, {q})) d{p}d{q} . (4.5) This is not possible to solve. The ergodic principle tells us we can replace this by a time average
hai = limN →∞ 1 N N X t=1 a(t) (4.6)
if the system is ergodic. That means the system should forget its initial state and travel through all parts of phase space. If we were to solve the equations of motion analytically for our molec-ular dynamics simulation the time evolution would be perfectly deterministic. Molecular dynamics exhibits a sensitivity to ini-tial conditions. In figure 4.1 we see the exponenini-tial divergence between two simulations that differ initially only by one part in
E R G O D I C I T Y A N D P H A S E S P A C E S A M P L I N G
FIGURE 4.1: Illustrations re-garding the Lyapunov instabil-ity in a Lennard-Jones system. The left panel shows the mean separation per atom vs time, h∆r(t)i = 1
N X
i
|ri(t) − ˜ri(t)|
between two systems. The sys-tems differ only in the addition of a random noise of one part in106to the initial coordinates.
It has a clear exponential be-haviour, and this exponent is called the Lyapunov expo-nent. The right panel shows the magnitude of this exponent as a function of system size. It is clear that the ergodicity imposed by this exponent de-creases drastically with system size. M ea n s ep ar at io n ( Å ) Sep ar at ion e xp on en t ( 10 -3 fs -1 )
Time (ps) 16 128 256 Number of atoms512 768 1024
3.7 3.8 3.9 4.0 4.1 4.2 4.3 4.4 4.5 4.6 0 0.5 1.0 1.5 2.0 2.5 3.0 10-7 10-6 10-5 10-4 10-3 10-2 10-1
10−6. This, together with other factors, ensures that the simula-tion forgets its initial condisimula-tion and can therefore be considered ergodic.
In classical molecular dynamics, often using millions or bil-lions of particles, ergodicity can be considered intrinsic. In BOMD, with particle numbers typically ranging from the tens to the hun-dreds, one can end up in situations where the system is not er-godic. If it happens the properties derived from the simulations are void of meaning. While it is difficult to explicitly measure er-godicity, we have developed some ideas to test the prerequisites of ergodicity.
From a simulation with Nttimesteps and Naatoms we can
construct the matrix
P= rx1(1) r1x(2) · · · r1x(Nt) r1 y(1) r1y(2) · · · r1y(Nt) .. . ... . .. ... rNa y (1) ryNa(2) · · · rNya(Nt) rNa z (1) rzNa(2) · · · rNza(Nt) p1x(1) p1x(2) · · · p1x(Nt) .. . ... . .. ... pNa x (1) pNxa(2) · · · pNxa(Nt) (4.7)
from the positions and momentums. The Ntrow vectors describe
the points in phase space we have visited. We then do a singular value decomposition
P= P ΣU. (4.8)
The diagonal matrix Σ holds the singular values. The effective rank of P is the number of nonzero entries in Σ. The effective rank is the number of dimensions of the phase space of the sim-ulation. If this number is not 6(Na− 1) we do not span phase
FIGURE 4.2: Normal mode in-tensities over time. The sys-tem was initiated in a single mode. This mode then decays in steps, to finally settle at the equilibrium distribution of modes. This example is for a Lennard-Jones crystal of 125 atoms in a face-centered cubic cell. The normal modes are or-dered by energy from right to left.
Normal mode
Time
Spanning and sampling phase space is not the same thing. But if one does not span phase space one cannot sample it. This sin-gular value decomposition is a routine check after a simulation has finished. If the spanning is not appropriate no thermal pa-rameter extracted holds any actual information.
If we have a harmonic potential for the system in question the phase space trajectories can be probed in a more direct way. By using the canonical transformation of equation 3.10 we can access the normal mode coordinates over time. In figure 4.2 the normal mode intensities are plotted over time. It is initialised to only oc-cupy a single mode. One can clearly see the decay of this mode into a steady state of thermally appropriate occupation. This is a good indication that the system is ergodic. Initiating the system in a single mode is not particularly useful in general. The coordi-nate transformation still provides insight: the presence of fluctu-ations in the intensities indicate finite phonon lifetimes, which is a requirement for ergodicity.
E N S E M B L E S A N D T H E R M O S T A T S
FIGURE 4.3: Trajectories for a classical two-dimensinal har-monic oscillator using different ensembles and thermostats. We see how the Nosé-Hoover thermostat slightly disturbs the trajectories and how the An-dersen thermostat is a series of microcanonical trajectories. The phase space sampling of the Langevin dynamics is supe-rior, and the trajectories are for the most part smooth with the occasional kink.
Ensembles and thermostats
Often the reason one turns to molecular dynamics is to see what effect the added thermal entropy has on some property at tem-perature T . Integrating the equations of motion as they stand yields the microcanonical ensemble, with constant number of par-ticles, volume and energy. Averaging over states to find a ther-mal average is cumbersome, as is controlling the temperature.
One would rather have the simulations in the canonical en-semble, constant number of particles, volume and temperature. In the macroscopic sense a constant temperature is easily imag-ined, one puts the system in contact with a large heat bath. In that case, the temperature is related to the time averaged kinetic energy as T = 3 2kB 1 Na− 1 Na X i Miv2i 2 . (4.9)
With a finite number of particles in a microcanonical ensemble the instantaneous temperature will fluctuate. We would like these fluctuations to be around the desired temperature T . To trans-form the microcanonical to the canonical ensemble with temper-ature T a thermostat is introduced.
It could be as simple as velocity scaling. Velocity scaling ad-justs the velocities every n timestep to match the kinetic energy to the desired temperature. This is not a particularly efficient technique. It does reproduce a canonical ensemble if the scaling is done at every timestep and in the limit of a large number of particles, something that can not be satisfied in ab initio molecu-lar dynamics. The trajectories will also be discontinuous at every rescaling step.50
Temperature of a system is defined through contact with a heat bath. The Andersen2thermostat introduces a probability at every timestep that an atom will collide with the heat reservoir, gaining a new momentum drawn from the appropriate Maxwell-Boltzmann distribution. This series of microcanonical ensembles will over time represent a canonical ensemble. For small systems the perturbation introduced by the atom with the new velocity can have undesirable effects. If the system consist of∼100 atoms
(I)Mode locking is a phe-nomenon occuring when the potential is harmonic or near harmonic. When a system is mode-locked the trajectories are uniquely defined by their initial conditions. It is well illustrated in figure 4.3: the microcanonical ensemble on a two-dimensional harmonic oscillator will yield a trajectory for the particle confined to the initial circle. The use of a stochastic thermostat is the only way to break this pattern.
it will take some time to recover to an equilibrium state. To rem-edy this the collision has to be set low, but rare collisions will make the time to reach a canonical ensemble very long.
The Nosé-Hoover49,33thermostat couples the equations of
mo-tion to an external system with fictitious mass and an extra de-gree of freedom that controls the temperature. Compared to the stochastic schemes it only weakly disrupts the trajectories. It does not break the mode-lockingI of near harmonic systems,33
which must be taken into account when, for example, perform-ing a thermodynamic integration. The accurate trajectories are well suited for small systems since only a smooth disturbance is introduced.
Another stochastic way of controlling temperature is to solve Langevins equations of motion instead of Newtons:
˙ri=
pi
Mi
˙pi= fi+ γpi+ ξ.
(4.10)
These equations were originally intended to describe Brownian motion, with a friction parameter γ and a random contribution ξ to the forces from collisions. These equations can be integrated in a Verlet-like scheme1 with the random collisions controlling the
temperature and speeding up the phase space sampling. In figure 4.3 we ran simulations using the different schemes for a two-dimensional harmonic oscillator. The mode-locking of the non-stochastic schemes is clearly illustrated.
Free energy and thermodynamic integration
Many parameters derived from molecular dynamics are mere indicators of what happens when the system is minimizing the free instead of the internal energy. High temperature elastic con-stants, averaged electronic densities of state or radial distribution functions are used to show that the system wants to go one way or another. If the full Gibbs free energy surface would be acces-sible most of these indicators would become redundant.
Unfortunately, The free energy is a thermal parameter and not directly accessible from the phase space trajectories. One can, however, extract it from a series of calculations coupling a har-monic and the ab initio potential energy surfaces.25
The partition function for a system with a given potential en-ergy function U ({r}) is
Z =Λ3N1N !
Z
exp (−βU({r})) d{r} (4.11) and we remember that Helmholz free energy is given by
F R E E E N E R G Y A N D T H E R M O D Y N A M I C I N T E G R A T I O N
The Kirkwood coupling technique38starts by writing U as a
lin-ear combination of two potential energy functions
U ({r}, λ) = (1 − λ)Ui({r}) + λUf({r}). (4.13)
Inserting this into the equations above and differentiating with respect to λ gives us ∂F ∂λ =− kBT ∂ ∂λlnZ(λ) =kBT Z(λ) ∂Z(λ) ∂λ = R ∂U ∂λexp (−βU({r, λ})) d{r} R exp (−βU({r, λ})) d{r} = ∂U ∂λ λ t =hUf − Uiit (4.14)
and the opportunity to integrate it Ff − Fi=
Z 1
0 hU
f − Uiitdλ. (4.15)
If we choose Uito be a harmonic potential energy we know
ex-actly the free energy of that system. We can then integrate to the free energy of any system, Ff. This is practically done by
performing molecular dynamics with the mixed Hamiltonian for several values of λ and integrating numerically.
T E M P E R A T U R E D E P E N D E N T
E F F E C T I V E P O T E N T I A L
Born–von Karman-theory describes the potential energy surface of the atoms as a Taylor expansion from the ideal positions at zero temperature. At finite temperature the potential energy surface can be significantly different. Molecular dynamics provides ac-curate trajectories along this finite temperature surface, but ther-modynamic properties are significantly harder to determine.
As mentioned in chapter 3 the idea of sampling the potential energy surface at a thermally excited state is not new.39It came to be when treating solid He, where the Taylor expansion fails to predict a stable state even at arbitrary order. In his 1955 pa-per32Hooton introduced the idea of an effective potential energy
surface that changes with temperature, and to predict thermal behaviour the surface should be sampled in its relevant states. Modern tools such as BOMD were not available at the time, and the work was focused on developing thorough formalisms. With modern DFT-based techniques the problem is significantly eas-ier. We can directly probe the potential energy surface and invert the problem using basic algebra.
I will first describe my approach, the temperature dependent effective potential method (TDEP), using a single anharmonic os-cillator in one dimension, to make it clear how straightforward the method is. I will then generalise it to the three dimensional periodic lattice, to follow up with examples and some details on the numerics.
A one-dimensional anharmonic oscillator
Consider the following one-dimensional potential: U (x) = k(x− x0)
2
2 + αe
−β(x−x0)2. (5.1)
It is a harmonic potential with an extra term to make it anhar-monic. The equilibrium position can depend on temperature and is shifted an unknown amount x0from 0, the assumed
equilib-rium position at T = 0. The aim is to find the second degree polynomial that best describes the system. If this polynomial only consist of a quadratic and a constant term it will describe a harmonic oscillator with well defined energy levels, making the extraction of thermodynamic properties trivial (see chapter 3). If one where to apply the harmonic approximation to potential 5.1 it would not work well. The second derivative
d2U dx2 = k− 2αβe −β(x−x0)2+ (4αβ(x − x0))2e−β(x−x0) 2 (5.2)
True potential TDEP Harmonic FIGURE 5.1: Comparison of per-formance of TDEP and the har-monic Taylor expansion for the potential described by equa-tion 5.1. Three examples when the harmonic approximation fails to describe the potential and TDEP succeeds. The left hand panels show the poten-tials and the right hand show the forces.αandx0are the
parameters in equation 5.1. In panel (a) the harmonic ap-proximation is dynamically un-stable, whereas TDEP provides an appropriate description. In panel (b) the harmonic approx-imation provides an inaccurate potential. Panel (c) shows how TDEP finds the high tempera-ture equilibrium position.
Displacement Fo rc e, a rb itr ar y u ni ts En er gy , a rb itr ar y u ni ts Displacement α<0 x0=0 α>0 x0=0 α>0 x0>0 -5 -4 -3 -2 -1 0 1 2 3 4 5 -5 -4 -3 -2 -1 0 1 2 3 4 5 (a) (b) (d) (c) (e) (f)
will determine the force constant Φ. Since we are unaware of the shift of the equilibrium from the ideal position at x = 0 we will end up with Φ = d 2U dx2 x=0 = k + 2αβ(2αβx0− 1)e−βx 2 0. (5.3)
This, as seen in figure 5.1, will not be a particularly good model for the true potential. These issues arise from the fact that the po-tential energy surface is only probed at the equilibrium positions. To work around this problem, let us put a particle in the poten-tial given by equation 5.1 and perform a MD simulation. When controlled by an appropriate thermostat, the particle will yield a set of Ntforces, positions and energies,{Ft, xt, Et}, one for each
timestep. This data can now be used to fit a potential of the form U (x) = ˜Φ(0)+1
2Φ˜
(2)(x
− x0)2, (5.4) a harmonic potential centered at x0. Let us begin by determining the second order term. To do that we guess a value for x0, use the forces from molecular dynamics{Ft} and minimize
min ˜ Φ(2)∆F = minΦ˜(2) 1 Nt Nt X t=1 |Ft− ˜Φ(2)(xt− x0)|. (5.5)
This is easiest realized as a least squares fit of a straight line in forces, as demonstrated in the right panel of figure 5.1. Equation 5.5 determines the second order term. The residual force at x0,
A O N E - D I M E N S I O N A L A N H A R M O N I C O S C I L L A T O R
FIGURE 5.2: Comparison of the effects of including higher or-der terms between the har-monic approximation and TDEP. When extending the tay-lor expansion of an anharmonic potential (dashed blue line) in the Born–von Karman ansatz to higher order terms we end up with the series of lines de-picted in panel (b). Panel (a) shows the same extension for TDEP. Even limiting the expan-sion to the second order term (n = 2), the fit will implicitly contain anharmonism to an arbitrary degree in the range that is thermally accessible. Extending TDEP to a higher order converges towards the true potential faster than when higher order terms are added to the harmonic approximation. Displacement En er gy n=6n=4 n=2 n=6 n=4 n=2 (a) (b)
∆F , can be used to find the equilibrium position x0. It is done in the following manner: a guess for x0gives us a ˜Φ(2)and ∆F . This residual force is used to move x0to a new position, and the pro-cess is repeated. When the equilibrium position is determined we can safely assume that any first order term in our polynomial can be set to zero. The constant energy term, Φ(0), can be deter-mined from the energies{Et}:
Φ(0)= Et− 1 2Φ˜ (2)(x t− x0)2 t . (5.6)
This is the best possible potential of the harmonic form at a given temperature. In figure 5.1 the true potential and the fit is illus-trated for different α,β and x0. The anharmonism of the potential is implicitly described by the polynomial fit. In figure 5.2 the ex-pansion has been extended to higher orders for an anharmonic potential. TDEP, probing the effective potential at finite temper-ature, converges to the true potential rapidly whereas including more terms in the Taylor expansion does by no means guarantee numerical stability at finite temperature.
(I)A note on the dimensions of the matrices on the last line of the equation: the column vectors of forces are flattened and have3Narows. The forces
are stored at each timestep, and both the force and dis-placement matrices will have Ntcolumns.
Extending the method to the periodic lattice
In order to describe a periodic lattice in a crystal potential the framework has to be generalized. The model potential in the one-dimensional case is replaced by DFT and the phase space is sam-pled using BOMD. This will provide positions, forces and ener-gies at finite temperature. Application of TDEP is just as straight-forward as the polynomial fit for the one dimensional case, but since the simulations can only be performed with limited accu-racy some effort must be put into handling the numerical errors. The starting point is to introduce a model Hamiltonian on the harmonic form for Naatoms:
ˆ H = U0+ Na X i=1 p2 i 2Mi + Na X i=1 Na X j=1 uiΦ¯¯ijuj (5.7)
where the interatomic force constants Φ and the ground state en-ergy U0are yet to be determined. In this model system the forces
are given by f1 f2 .. . fNa | {z } FH t = ¯ ¯ Φ11 Φ¯¯12 · · · Φ¯¯1Na ¯ ¯ Φ21 Φ¯¯22 · · · Φ¯¯2Na .. . ... . .. ... ¯ ¯ ΦNa1 Φ¯¯Na2 · · · Φ¯¯NaNa | {z } ¯ ¯ Φ u1 u2 .. . uNa | {z } Ut . (5.8)
This is a reformulation of equation 3.9 to matrix form. The map-ping between this form and the one used in chapter 3 is deter-mined by the BOMD simulation cell. What we seek is the effec-tive second order force constant matrix ¯Φ. A BOMD simulation¯ will provide a set of displacements{UMD
t }, forces {FMDt } and
po-tential energies{EMD
t }. We seek to minimize the difference in
forces from BOMD (FMD) and our harmonic form (FH):I
min ¯ ¯ Φ ∆F = min ¯ ¯ Φ 1 Nt Nt X t=1 F MD t − FHt 2 = = min ¯ ¯ Φ 1 Nt Nt X t=1 F MD t − ¯ΦU¯ MDt 2 = = min ¯ ¯ Φ 1 Nt F MD 1 . . . FMDNt − ¯¯Φ U MD 1 . . . UMDNt . (5.9)
This is realized with a a Moore-Penrose pseudoinverse ¯ ¯ Φ = FMD 1 . . . FMDNt UMD 1 . . . UMDNt + (5.10) to obtain the linear least squares solution for ¯Φ. This is then¯ mapped to the form
¯ ¯
E X T E N D I N G T H E M E T H O D T O T H E P E R I O D I C L A T T I C E
FIGURE 5.3: Illustration of the symmetry equivalent inter-actions. Presented is a two dimensional lattice with two species of atoms interacting with their nearest and next nearest neighbours. After con-sidering the symmetry of the lattice we find that there are just three inequivalent interac-tions. The differently coloured arrows represent these irre-ducible interactions.
where αβ are indices to a unit cell with Nuc≤ Naatoms and µν
are Cartesian indices. The pair vectors in the supercell Rij are
mapped to stars of vectors Rlconnecting atoms of type α and β.
From this form the phonon dispersion relations, free energy and all other quantities are extracted. This direct implementation of TDEP works well, but the numerical efficiency can be improved. In figure 5.3 we have a diatomic system where we consider interactions up to the second nearest neighbour. The straightfor-ward implementation described above would solve for Φ with all the (3Na)2components of Φ as unknown quantities, but
judg-ing from the figure there should be several interactions that are equivalent. Each arrow in the figure represent a force constant, and there are due to symmetry only three inequivalent interac-tions. The following section describes how to constrain the ex-traction of the interactions making them obey the proper sym-metry.
Symmetry constrained force constant extraction
The form of the force constant matrices depends only on the su-percell used in the BOMD and the crystal lattice. We begin by populating the force constant matrices Φαβ
µν(Rl) with unknown variables θk ¯ ¯ Φ11(R1) = θ1 θ2 θ3 θ4 θ5 θ6 θ7 θ8 θ9 ¯ ¯ Φ11(R2) = θ10 θ11 θ12 θ13 θ14 θ15 θ16 θ17 θ18 (5.12)
and so on, including vectors Rlup to a cutoff determined by the
size of the supercell. Many of these θkare equivalent. To find
out which, we study the symmetry relations the force constant matrices have to obey. We have
X lα ¯ ¯ Φαβ(Rl) = 0 for each β (5.13) ¯ ¯ Φαβ(Rl) = ¯Φ¯ βα (−Rl) (5.14) Φαβµν(Rl) = Φαβνµ(Rl) (5.15)
that stem, in order, from the facts that there is no net translation of the crystal, all Bravais lattices have inversion symmetry, and that the order of the second derivatives does not matter. Each relation will give a few equations for the unknowns θk, reducing
the number of independent variables. In addition to these fun-damental properties of the force constant matrices we can ben-efit from the symmetry of the lattice. If symmetry operation S, belonging to the point group of the lattice, relates two vectors Rl= SRkwe have the following relation:44
¯ ¯
Φαβ(Rl) = S ¯Φ¯ αβ
(Rk)ST. (5.16)
By applying equations 5.13–5.16 the number of inequivalent θk
are massively reduced. For example, a body centered cubic (bcc) lattice modelled as a 4× 4 × 4 supercell (128 atoms) has 147456 unknowns in ¯Φ not considering symmetry. The symmetry re-¯ duced problem has 11 unknown variables θk. Having found the
reduced problem with Nθ unknown variables, it can be
S Y M M E T R Y C O N S T R A I N E D F O R C E C O N S T A N T E X T R A C T I O N
(II)Once again a note on the dimensions: the force vector will have3NaNtrows. Each
submatrixC¯¯ will have3Na
rows andNθcolumns. The full
coefficient matrix has3NaNt
rows andNθcolumns. Θis a
row vector withNθrows.
schematically look like: f1 f2 .. . fγ .. . f3Na | {z } FHt = θ1 0 0 . . . 0 θ1 0 . . . 0 0 θ1 . . . θ2 θ3 −θ4 . . . θ3 −θ2 0 . . . −θ4 0 0 . . . .. . ... ... . .. | {z } ¯ ¯ Φ u1 u2 .. . uδ .. . u3Na | {z } Ut . (5.17)
The actual distribution of the θkwill depend on the problem at
hand. Carrying out the matrix product produces a new set of equations for the forces
fγ= X k θk X δ ckγδuδ, (5.18)
where second sum describes the coefficients for each θkcontained
in the expression for force component fγ. These coefficients are
linear combinations of the displacement components uδ. The
ex-plicit form is determined by the lattice. In matrix form this is written
F = ¯C(U)Θ,¯ C(U)kγ=
X
δ
ckγδuδ. (5.19)
This equation is equivalent to equation 5.8, it is just rewritten in the terms of the symmetry inequivalent interactions. Our im-plementation of TDEP symbolically reduces the number of un-knowns, generates the function that gives the matrix ¯C from a¯ set of displacements U and the mapping from the set of θkback
to the force constant matrix Φαβ µν(Rl).
To find Θ we repeat the process used to find Φ: minimize the difference in forces between BOMD and the model Hamiltonian
min Θ ∆F = minΘ 1 Nt Nt X t=1 F MD t − FHt 2 = = min Θ 1 Nt Nt X t=1 F MD t − ¯C(U¯ MDt )Θ 2 = = min Θ 1 Nt FMD 1 .. . FMD Nt − ¯ ¯ C(UMD 1 ) .. . ¯ ¯ C(UMD Nt) Θ . (5.20)
This is again realised with the least squares solution for ΘII
Θ = ¯ ¯ C(UMD 1 ) .. . ¯ ¯ C(UMD Nt) + FMD 1 .. . FMD Nt . (5.21)
Having determined Θ we can substitute the components into the force constant matrix and proceed to calculate phonon dispersion relations and thermodynamic properties.
This is a numerically superior way of using symmetry to im-prove the numerical accuracy. Most schemes involving symme-try revolve around determining the interaction in one direction and then using the symmetry relations to translate and rotate that interaction. The unavoidable numerical errors will propagate to all interactions, whereas in my method the errors will be aver-aged and should cancel each other out.
I N T E R N A L D E G R E E S O F F R E E D O M
(III)This scheme ensures that the first order term in
U =U0+ X iα Φαiuαi+ 1 2! X ijαβ Φαβij uαiuβj + . . .
can be safely set to zero. If that is not the case, the equations of motion can not be solved with a plane wave ansatz and the phonon formalism breaks down. The positions that minimize the residual forces are the positions where the derivative of energy with respect to displacement should be zero, that is they are the equilibrium positions.
Internal degrees of freedom
Just as in the one-dimensional case we have started with deter-mining the second order force constants. If the system has ternal degrees of freedom, such as a system with defects, an in-terface or a random alloy, the atoms ideal positions and equilib-rium positions do not coincide. While one could find the relaxed positions from 0K calculations, the equilibrium positions are by no means constant with respect to temperature. TDEP handles this in an elegant fashion. Just as for the one-dimensional os-cillator we find the second order terms with a least squares fit of the slope of force versus displacement. Even though the dis-placements are calculated with respect to the wrong equilibrium position the slope will be the same. That allows for the following procedure:III
– Guess equilibrium positions, usually the ideal lattice posi-tions.
– Use these to calculate the displacements u from the BOMD simulations.
– Determine Θ and from that the residual force Fr= Nt X t=1 1 Nt FMDt − FHt, (5.22)
where FMDt are the BOMD forces and FHt are given by
equa-tion 5.19.
– Use forces are then used to move the atoms towards the equilibrium positions.
– Repeat until convergence.
Tests shows that this procedure is remarkably stable. The sec-ond order force constants Φ are weakly affected by the choice of equilibrium position. The vibrational entropy and phonon dispersion relations are largely unaffected as well. Eliminating the first order term, however, is formally important and crucial when extracting higher order terms. In figure 5.4 the difference between the ideal positions, the 0 K relaxed positions and the finite temperature positions of reduced ceria is illustrated.
