of the optial properties
of phase-hange materials
Manfred Niesert
Version: August 11, 2003
1 Introdution 1
2 Density Funtional Theory 3
2.1 The Many-Partile Problem . . . 3
2.2 The Hartree-Fok Ansatz . . . 4
2.3 Density Funtional Theory . . . 5
2.4 Exhange and Correlation . . . 8
2.5 Notes onSymmetry . . . 9
2.6 Spin-Density Funtional Theory . . . 11
2.7 Determination of the TotalEnergy . . . 11
2.8 Improvements to Density Funtional Theory . . . 12
3 Dieletri Properties of Solids 15 3.1 Physial Tensors . . . 15
3.2 Marosopi Optis . . . 16
3.3 Relation between real and imaginarypart . . . 20
3.4 Ideas . . . 21
3.5 Transmissionspetra . . . 21
3.6 Classial Models . . . 22
3.7 Dieletri funtion . . . 22
3.7.1 Mirosopi denition, dieletriity . . . 22
3.7.2 Fouriertransforms . . . 23
3.7.3 dieletriity $ internal harge density . . . 24
3.8 Quantum mehanialmodel . . . 24
3.9 Comparison . . . 24
3.10 Missing . . . 25
3.11 Krasovskii . . . 25
4 Basis sets 27 4.1 The Plane-Wave Basis . . . 27
4.2 The APW method . . . 29
4.3 The LAPW method. . . 32
4.4 The LoalOrbitalextension . . . 33
4.5 Notes on symmetry . . . 34
4.6 Notes on the kineti energy operator . . . 34
5 Implementation 35 5.1 Momentum matrix elementsin the LAPW basis . . . 35
5.1.1 Interstitialontribution. . . 36
5.1.2 MuÆn-tinontributions . . . 37
5.1.3 Properties of the matrix elements . . . 38
5.1.4 Illustration . . . 39
5.2 k-spaeintegration . . . 40
5.3 The Realpart of the Dieletri Funtion . . . 46
5.4 Bakfolding . . . 47
5.4.1 Algebrai onsiderations . . . 47
5.4.2 Representation ina basis . . . 50
5.4.3 Illustration . . . 54
5.4.4 Consequenes for omputation . . . 56
5.5 Resultingproblems inthe integration . . . 58
5.5.1 The Inuene of Degeneray . . . 58
5.5.2 The Inuene of Bandrossing . . . 60
5.6 A Noteon ComputationalDemands . . . 62
5.7 Test alulation . . . 64
5.7.1 Aluminum . . . 64
5.7.2 Copper. . . 66
6 Results 69 6.1 GeTeompounds . . . 69
6.2 Calopyrites . . . 70
7 Conlusion 71 A Momentum matrix elements 73 B Tetrahedron method 79 B.1 Integration weights . . . 79
B.2 Number and density ofstates . . . 81
C Units 83
D Parameters of alulations 85
Bibliography 90
Danksagung 91
Introdution
Density Funtional Theory
2.1 The Many-Partile Problem
Theompletepropertiesofsolidsaninpriniplebealulatedabinitio {i.e. free
ofany parameters, onlyusingthe setup ofthe system and itsinterations {ona
quantummehaniallevel. Thewholeinformationofasystemisontainedinthe
system's wave-funtion,whih has to be obtained as solutionof the Shrodinger
equation
H j i=Ej i; (2.1)
withH theHamiltonianofasystemofinterating nuleiandeletrons(assuming
4"
0
=1)
H= N
X
i=1
~ 2
2m r
2
i +
1
2 X
i6=j e
2
jr
i r
j j
X
i;J Z
J e
2
jr
i
J j
+ 1
2 X
I6=J Z
I Z
J e
2
j
I
J j
: (2.2)
r denote the eletronial oordinates and those of the nulei, Z
I
denotes the
harge of the nulei. Spin-dependene and external elds are omitted. In the
relativistiase, the Dira equation has tobesolved. The energy of a state is
given by
E =h jH j i: (2.3)
Theeorttosolvethismany-bodyproblemsalesexponentiallywiththenumber
of partiles desribed and is unaomplishable for everything exept very small
systems,and ertainlyforamarosopisystem with anumberofpartilesof an
order ofmagnitude of 10 23
.
ArstandverygeneralapproximationistheBorn-Oppenheimermethod(also
alledadiabatiapproximation). Sinethe mass of the eletrons isatleast three
orders of magnitude smaller than those of the nulei, the eletrons are expeted
to follow the motions of the nulei instantaneously, while the nulei will reat
slowly to a hange in eletroni onguration. Therefore, the ion's position an
be set xed, reduing the number of degrees of freedom. (From a strit point
of view this approximation needs more preise justiation, see [Mad78℄.) This
approximationis used inthe majorityof eletroni alulations.
When alulating the ground state of a system, the energy has to take its
minimum. Depending on your ansatz, the solution an usually be obtained by
minimizingthe total energy.
2.2 The Hartree-Fok Ansatz
Avarietyofdierentapproaheshavebeendevelopedtotaklethismany-partile
problem. One frequently used method (in many areas of physis) is to transfer
the many-bodyproblemto aone-partile-like problem,for instane by imposing
some ertain formonthe wavefuntion.
The mostbasi hoieis theHartree Ansatz, whihreplaes the wavefuntion
(r
1
;:::;r
N
)with a produt of N one-partilewavefuntions (r):
(r
1
;:::;r
N )=
1 (r
1 )
2 (r
2
):::
N (r
N
); (2.4)
depending only on the spatial oordinate of one partile. If one introdues
this ansatz intothe Shrodinger equation, one obtains N Shodinger-like single-
partile equations with a integral alled Coulomb term or Hartree term, on-
taining the eletron-eletron interation. This simple ansatz treats the partiles
independent in the sense that every partile moves in a stati potential reated
by the other eletrons, whih is the only interation onsidered.
It is possible to take are about the expelling properties of fermions result-
ing from the Fermi priniple { alled exhange interation { by using a slater
determinantof wavefuntions instead of asimple produt:
(r
1
;:::;r
N )=
1
p
N
1
(r
1 ) :::
N (r
1 )
.
.
. .
.
. .
.
.
1 (r
N ) :::
N (r
N )
: (2.5)
ThisHartree-Fok Ansatzresultsinasigniantlymoreomplexnumerialtreat-
mentaswellasinmuhbetterresults. Inludingawavefuntionofthis forminto
the ShrodingerequationgivesN singlepartileequationsnowontaininganad-
ditional term { the exhange or Fok term { ontaining ontributions from all
the other single-partilewavefuntions.
The desriptionisstillinompletedue tothefatthat thesingle partilesare
not independent asassumed inthis approximation. Theseorrelation eets an
2.3 Density Funtional Theory
A new idea how to desribe the ground state of a many-partile system has
been aquiredbyHohenberg and Kohninthe 1960s. It turns the fousfrom the
abstratmany-partilestateasdesriptivequantityofthesystemtotheostensive
hargedensity inreal spae. Notonly thatnot the whole informationontent of
thewave-funtionisneeded, itisnot desirabletoobtainthe omplete solution
fora large system sine storage of itis as hardly possible asalulation of it.
One dierent approah, the Thomas-Fermi theory, was known sine the late
twenties [Fer27,Tho27℄. It assumes the interating eletrons to be independent,
movinginanexternal potential. (In this ontext theterm external meansevery-
thingexept of this one partileitself, soitinludes alsothe eets of the nulei
inthesystem, notonlythoseofeldsexternaltothesystem.) Thentheformulae
for the uniform eletron gas are applied. The obtained results give only a few
quantitativetrends,hemialsbondsfor instaneannot bepredited. However,
the system is desribed by the density only.
The Lemma of Hohenberg and Kohn: The harge density relates to the
many-partile wavefuntion like
n(r)=
j N
P
i=1
Æ(r r
i )j
: (2.6)
The amount of harge
N = Z
d 3
r n(r) (2.7)
takes the role of a subsidiaryparameter.
Hohenberg and Kohn derived that the expetation value of any observable
is uniquely dened by the harge density. Furthermore, the funtional of total
energy
E =E[n(r)℄ (2.8)
isminimizedby the true groundstate density n
0
(r). As a third point,the
The important onlusion of the Lemma of Hohenberg and Kohn [HK64℄ is
that the density n(r) of the ground state of a system of interating eletrons in
some external potentialv(r) determines this potentialuniquely (of ourse up to
someunimportantonstant). The proofis shown inontraditionfor the energy
funtional
E
g
=h
g jH j
g
i (2.9)
of a non-degenerate ground state, whih is shown to be expressable in terms of
the density,
E =E[n(r)℄: (2.10)
It is shown that It an easily be extended to the degenerate ase [Koh85℄.
This means that n(r), determining the potential v(r) and the number of
partilesN,desribestheHamiltonianandtherewiththeompletesystemandall
itsderivableproperties(inludingmany-bodywavefuntions,two-partileGreen's
funtions). A more mathematial insight is that there are funtions n(r) not
yielding a valid potential v(r), so-alled non V-representable funtions. These
are non-physialdensities.
The Hohenberg-Kohn lemma doesnot implyany knowledge about the phys-
ial interations and is universal thereby. On the other hand, nothing has been
stated about the form of the funtional E[n℄ up to now.
Kohn-Sham equations: Kohn and Sham formulated a form for the energy
funtional that proved to be very suessful. They proposed to split it up into
three ontributions
E[n℄=T
s
[n℄+U[n℄+E
x
[n℄: (2.11)
T
s
is the kineti energy of non-interating partiles, U is the Coulomb energy,
and E
x
ontains the remaining ontributions to the energy due to exhange
and orrelation. The Coulomb energy of the eletrons is onstruted out of the
eletron-eletron energy together with the external energy, resulting additively
from the Coulomb eld of the nulei and from elds externalto the system:
U[n℄ = E
ext
[n℄+E
H
[n℄ (2.12)
E
ext [n℄ =
Z
d 3
r V
ext
(r)n(r) (2.13)
E
H
[n℄ = e
2
8"
0 Z
d 3
rd 3
r 0
n(r)n(r 0
)
jr r 0
j
(2.14)
An advantage of this representation is that for the kineti energy, whih is a
signiant proportion to the total energy, an analyti expression an be given
(see setion 2.7). The density isrelated tothe singlepartile wavefuntions via
n(r)=2 N
X
i=1 j
i (r)j
2
; (2.15)
with the fator 2 aounting the spin degeneray 1
. For this hoie the kineti
energy reads
T
s
[n℄= 2 N
X
i=1 Z
d 3
r
i (r)
~
2m r
2
i
(r): (2.16)
1
Equivalenttominimizingtheenergywithrespet tothedensity,oneandosoas
wellwithrespettothesinglewavefuntionsortotheiromplexonjugates. The
subsidiaryonditionofpartileonservation(2.7)isreplaedbythenormalization
of the wavefuntions
Z
d 3
r j
i (r)j
2
=1: (2.17)
Takingthis requirementintoaountbyLagrangeparameters
i
, thevariationof
the energy yieldsthe Kohn-Shamequations
H
1 i (r)=
~
2m r
2
+V
eff (r)
i
(r)=
i i
(r); (2.18)
whihareShrodinger-likeequationsofaone-partileHamiltonianH
1
ontaining
aneetive potential
V
eff
(r)=V
ext
(r)+V
H
(r)+V
x
(r) (2.19)
onsisting of the external, the Hartree and the exhange-orrelation potential
V
ext
(r) = Æ
Æn(r) E
ext
(r) (2.20)
V
H
(r) = 4e 2
Z
d 3
r n(r
0
)
jr r 0
j
(2.21)
V
x
(r) = Æ
Æn(r) E
x
(r): (2.22)
These potentialsare simple funtions, while the orresponding energies are on-
sideredas funtionalsof the density.
This hoie (2.11) of kineti energy and subsequent derivations onverts the
problemtoaproblemoftitioussinglepartilesmovinginaneetivepotential
allother partilesontribute to.
Theparameter
i
areintroduedasLagrangianparametersonly. Aordingto
Janak's theorem, onlythe highest oupied value has a physialmeaning, i.e.it
is equal to the hemial potential, the ionisation energy of the system. Beyond
this,thereisnojustiationtotaketheseparametersastheone-partileenergies.
However, it is known from experiene that this assumption works surprisingly
good,and this identity isommonly assumed in bandstruture alulations.
Eigenvalueproblem: Usuallythe Kohn-Shamequations(2.18)arenotsolved
diretly, but the solutions are represented in a basis. Then the operator H
1 has
tobeonstruted and diagonalized. Sinethe basis funtions are notneessarily
orthogonal,one has tosolvethe generalizedeigenvalue problem
(H
1
i
S)=0 (2.23)
(also alled seular equation) with S the overlap matrix and the expansion
Self-onsisteny: SinetheeletrondensitygoesintotheHartreepotentialV
H
andtheexhange-orrelationpotentialV
x
,andtheeetivepotentialdetermines
the solutions
i
through (2.18),whih againmake theharge density (2.15),this
formalism omprises a self-onsisteny, as shown in gure(2.1).
Figure 2.1: The self-onsisteny yleof adensity-funtionalalulation.
To enter the loop one has to provide an appropriate starting density. With
this thepotentialsare generatedand theone-partilesolutionsarealulated. In
matrix piture this is the setup of the H and S matriesand the solutionof the
generalized eigenvalue problem (2.23). With the results the temporary density
n
new
(r) is alulated.
One now heks if the dierene between the previous density n (i)
(r) and
the new one is suÆiently small. If not, the temporary density is inorporated
into the previous one. Sine taking the alulated density as next input density
n (i+1)
(r) for the yle would introdue too big steps whih destroy onvergene,
some mixinghas tobe performed. The simplestway is alinear mixing
n (i+1)
(r)=(1 )n (i)
(r)+ n
new
(r) (2.24)
with mixing parameter . More sophistiated methods like those of Broyden
and Anderson have been developed, whih inorporate the knowledge of earlier
iterationsandyieldafasteronvergene. Afternishingtheloop,oneanproess
the obtained density, e.g. alulatethe total energy.
2.4 Exhange and Correlation
Sine no approximations have been made so far, density funtional theory is
exat inpriniple. However, alulationsare onlypossiblewiththe knowledge of
the exhange-orrelation energy funtional E [n℄ dened by (2.11). The exat
funtionalisunknownandnotsolubleanalytially. Solvingitwouldbeequivalent
tosolving the many-body problem. Therefore, approximationshaveto bemade.
Basially, the Kohn-Sham equations are a Hartree-like ansatz. All exhange
and orrelationeets (i.e. allmany-bodyeets) are inluded inthe funtional
E
x
[n℄. It ontains the fermioni eets, modiations to the eetive potential
andorretions tothe kinetienergy,alldue tothe eletron-eletroninteration.
This means that the exhange-orrelation potential desribes the eets of the
PauliprinipleandtheCoulombpotentialbeyondapureeletrostatiinteration
of the eletrons.
The most widely used approah is the Loal Density Approximation (LDA).
The idea is toassume E
x
to be that of a homogenouseletron gas with density
n(r):
E
x
[n(r)℄= Z
d 3
r n(r)
x
(n(r)): (2.25)
The importantsimpliation is that
x
is not a funtionalof the density, but a
funtion of the value of the density at some spatial oordinate. With this, also
theexhange-orrelationpotentialV
x
in(2.20)takestheformofafuntion. One
possible approximation isto viewexhange and orrelation to be independent:
x
(n(r))=
x
(n(r))+
(n(r)) (2.26)
Moreomplexparametrisationsinorporatetheresultsof Hartree-Fok ormany-
bodyalulations. OnewouldexpettheLDAtofailsystemswithrapidlyvarying
densities. Butit shows to give goodresults inan unexpeted variety of systems.
A lass of more sophistiated approximations is the Generalized Gradient
Approximation (GGA). It makes the same loalization ansatz as in (2.25), but
onnets
x
not only with the value of the density but also with the absolute
value of itsgradient:
E
x
[n(r)℄= Z
d 3
r n(r)
x
(n(r);jrn(r)j): (2.27)
2.5 Notes on Symmetry
Symmetriesareoperationsthattransferasystemintoitself,sothatbothsystems
are indistinguishable. In this ontext we are interested in symmetry operations
inreal spae. Symmetry operators ommute with the Hamiltonian,
[( ;T);H ℄=0: (2.28)
( ;T) denotes anoperationonsisting ofa rotation and a subsequent transla-
tion T. Takingsymmetriesinto aountan massively simplifythe alulations,
Classiations: Perfet rystals, that are systems possessing translational
symmetry, are lassied into lattie types. Considering translations only (no
omplex oupations of the unit ell with atoms), this gives the minimal set of
essentiallydierent lattie types, the Bravaislatties. In three dimensions there
are 14 Bravais latties: the seven latties ubi, trigonal, rhombi, hexagonal,
monolini,triliniand tetragonal, dened by the length ofand angles between
the basis vetors, and variationsof these latties by oupying unit ellfaes or
theunit ellenter withatoms. Theaordingtranslationaloperatorsofalattie
form the Translationgroup.
The rotationsofasystem(i.e. theaording operators)thatbringtherystal
into itselfbuild the Rotation group. There are alsonon-symmorphi symmetries
whih bring the rystal into itself only with an additionaltranslation (whih is
not part of the translational group). The aording symmetry operations are
srew axis and glide planes. In this ase these rotations extend the rotation
group to the Point group. (For symmorphi latties both are idential.) There
are thirty-twodierent point groups.
The Spae group onsists of the totality of transformations that bring the
rystal intoitself,ontainingthetranslationaland thepointgroupassubgroups.
There are 230 possible spae groups; 157 of them are non-symmorphi, 73 are
simple.
Translational symmetry: The translational operator
T
R
: r!r+R (2.29)
for a lattie vetor R ommutes with the Hamiltonian. So both operators share
a set of eigenvalues. The onsequene is the so-alled Bloh theorem, that states
that the wavefuntions an takethe form
n
(k;r)='
n (k;r)e
ikr
; (2.30)
dening k(often alledtherystal momentum)asanew goodquantum number.
Thisvetorkistakenfromthereiproalspae,butoneanredueonsiderations
to the rst Brillouin zone. The spetrum of energy eigenvalues is periodi in
reiproal spae,
E(k) =E(k+G); (2.31)
G being a reiproallattie vetor.
Rotational symmetry: Toarotationinreal spae,the aording symmetry
operationinthereiproalspaeistheinverserotation. Analogouslytothetrans-
lations, this redues the eetive reiproal spae you have to onsider, leaving
as unique part the irreduible wedge of the rst Brillouinzone(IBZ).
2.6 Spin-Density Funtional Theory
The spin property of eletrons, so far only aounted by a degeneray fator of
two, an be easily inorporated into the theory. It has been shown that the
basi Hohenberg-Kohn theorem stands for spin-polarized densities as well. You
redene (inthe non-relativistiase) the wavefuntions as spinors
i (r)=
i"
(r)
i#
(r)
!
: (2.32)
With this slightly dierent notation, apart from the harge density there arises
a seond entral quantity out of these wavefuntions, the magnetizationdensity
m(r):
n(r) = N
X
i=1 i
(r)
i
(r) (2.33)
m(r) = N
X
i=1 i
(r)
i
(r): (2.34)
is the vetor (
x
;
y
;
z
) of Paulimatries. The energy is now a funtional of
these two densities:
E =E[n(r);m(r)℄ (2.35)
Thetwospinsouplethroughaneetivemagnetieldappearinginthemodied
Kohn-Sham equations. To inorporate the interation of an external magneti
eld B
ext
with this spin-polarized system, we inlude the energy ontribution
m(r)B
ext
(r) intothe Kohn-Shamequations and yield
H
1 i
(r) =
~
2m r
2
+V
eff
(r)+B
eff (r)
i
(r)=
i i
(r); (2.36)
B
eff
(r) = B
x
(r)+B
ext
(r); (2.37)
B
x
(r) =
ÆE[n(r);m(r)℄
Æm(r)
: (2.38)
The approximations in setion 2.4 an be easily extended for the ase of spin-
polarizedsystems.
2.7 Determination of the Total Energy
When the total energy needs to be alulated, the ion-ion interation E
ii
of the
nulei
E
ii
=e 2
X
Z
I Z
J
j
I
J j
(2.39)
has tobe inluded into the funtional(2.11),
E
tot
[n℄=T
s
[n℄+E
H
[n℄+E
x
[n℄+E
ext +E
ii
: (2.40)
Beause of numerial reasons, it is not desirable to alulate the kineti energy
in the form (2.16), applying the double spatial derivative. Instead, one utilizes
the Kohn-Shamequations(2.18). Rearranging,multiplyingtheBra fromthe left
and summing overall oupied states gives
~
2m r
2
i
(r) = (
i V
eff (r))
i
(r); (2.41)
T
s
[n℄ = 2 N
X
i=1 Z
d 3
r
i (r)
~
2m r
2
i
(r) (2.42)
= N
X
i=1
i Z
d 3
r n(r)V
eff
(r) (2.43)
Putting allthe ontributions together weobtain
E[n;m℄ = N
X
i=1
i Z
d 3
r n(r)V
eff
(r) (2.44)
Z
d 3
r m(r)B
eff
(r) (2.45)
4e 2
M
X
I=1 Z
d 3
r
n(r)Z
I
jr
I j
(2.46)
Z
d 3
r n(r)
~
V
ext
(r) (2.47)
+4e 2
1
2 Z
d 3
rd 3
r 0
n(r)n(r 0
)
jr r 0
j
(2.48)
+ Z
d 3
r n(r)
x
(n(r);jm(r)j) (2.49)
+4e 2
M
X
I6=J Z
I Z
J
j
I
J j
; (2.50)
with the potential
~
V
ext
(r) due to aneletrield externalto the system.
2.8 Improvements to Density Funtional The-
ory
Many extensions has been madeto thedensity funtionaltheory,and it isstilla
the inlusion of external eletri and magneti elds are a natural extension of
the theory.
New exhange-orrelation funtionals are being developed. Methods like the
simple sissors operator or the more sophistiated LDA+U theory fous on one
of the entral drawbaks of the loal density (LDA) or generalized gradient ap-
proximation (GGA), the mismathing band-gap. The time-dependent density
funtionaltheory renes the knowledge about the development of the system in
time,and resultsin a better desription of exited states.
The density funtion theory has proven tobea very powerful tool totreat a
many-body problem eÆiently and preisely in the framework of a one-partile
piture. It has been applied also in a diversity of other disiplines, like super-
ondutivity orastrophysis.
Dieletri Properties of Solids
Beforegoing intothe detailsof the dieletri funtion,letusrst disuss general
properties of physial tensors(of rank two).
3.1 Physial Tensors
Letus onnet onnet two physial vetor quantities linearly via
B=Ta: (3.1)
IfBissimplyproportionaltoa(i.e. pointinginthe samediretion)T isasalar
fator. Butin the generalase, T is atensor of seondrank. Byits denition,a
tensortransforms under abasis hangeA to
T 0
=ATA T
; or T 0
ij
=A
ik A
jl T
kl
: (3.2)
Any seond-rank tensor an be split up into a symmetri and an antisymmetri
part,
T
ij
= 1
2 (T
ij +T
ji )+
1
2 (T
ij T
ji
); (3.3)
but most physial seond-rank tensors are purely symmetrial (i.e. T
ij
= T
ji ),
for example the dieletri tensor being subjet of this thesis. (One of the few
exeptionsisthethermoeletritensor.) Nye[Nye57 ℄remarksthatthissymmetry
propertyoftensorsisnotanobviousone,andthattheproofneessararilyinvolves
thermodynamialonsiderations.
Thebehaviourof asymmetriseond-rank tensorT
ij
under oordinatetrans-
formationis the same asfor the equation
T
ij x
i x
j
=1; (3.4)
whih denes a sphere that is either an ellipsoid, a hyperboloid of one or a
hyperboloid of two sheets. This equationis alled the representation quadri for
the tensor T
ij
. An important property of a quadri isthe possessionof prinipal
axes. These are three diretions at right angles suh that the general quadri
(3.4) takes the form
T
11 x
2
1 +T
22 x
2
2 +T
33 x
2
3
=1; (3.5)
when referred tothese axes.
In asymmetrialtensor referredtoarbitraryaxesthenumberofindependent
omponents is six. How many independent oeÆients remainwhen referring to
its prinipalaxes depends on the symmetry of the rystal in onsideration. The
Neumann priniple states that the symmetry elements of any physial property
of a rystal must inlude the symmetry elements of the pointgroup of the rys-
tal. As a result of these onsiderations, one groups the tensors (or the rystals,
aordingly) in the following three so-alledoptiallassiations:
Isotropi (Anaxial) rystals: Crystals in whih you an hoose arbitrarily
three rystallographially equivalent orthogonal axes. These three axes are the
prinipal axes of the tensor. All diagonal elements are equal (see table below),
and the rystall ats like anamorphous medium.
Uniaxialtensors: Crystalswithoutthreeorthogonalequivalentaxes, butwith
twoormoretheseaxesinoneplane. This isthe aseforthetriline,trigonaland
hexagonal latties. The plane with the equivalent axes is perpendiular to the
three-fold, four-fold or six-fold symmetry axis, respetively. One of the optial
axesoinideswiththissymmetryaxes, theothersformapairoforthogonalaxes
in the plane.
Biaxial tensors: Crystals with lower symmetry. For orthorombi rystals,
the tensorpossessesdiagonalformwitheahdierentelements. Theoptialaxes
oinidewiththerystalaxes. Inmonolineandtrilinesystems,theoptialaxes
are notalleged. (Inthisase, itwould bepossibletorotatethe axesofthe tensor
suh that only the three prinipal oeÆientsare neessary, but one would have
no information regarding the orientation of the representation's sphere relative
to the rystallographiaxes [Lov89℄.)
The orrespondingshape of the tensors istaken from atable of ([Nye57℄).
In most ases of alulations the used basis vetors oinide with the optial
axes of the rystal in study.
3.2 Marosopi Optis
Wemakeamarosopiapproahtothe eletromagnetidesriptionofamatter.
Classiation Crystal System Indep. Coe. Tensor shape
Anaxial Cubi 1
0
B
T 0 0
0 T 0
0 0 T 1
C
A
Uniaxial
Tetragonal
Hexagonal
Trigonal
2
0
B
T
1
0 0
0 T
1 0
0 0 T
3 1
C
A
Orthorhombi 3
0
B
T
1
0 0
0 T
2 0
0 0 T
3 1
C
A
Biaxial Monolini 4
0
B
T
11
0 T
31
0 T
2 0
T
31
0 T
33 1
C
A
Trilini 6
0
B
T
11 T
12 T
31
T
12 T
22 T
23
T
31 T
23 T
33 1
C
A
Table 3.1: Shapes of seond-rank tensors for dierent rystal strutures.
Maxwell equations:
rE(r;t) =
t
B(r;t) (3.6)
rD(r;t) = (r;t) (3.7)
rH(r;t) = j(r;t)+
t
D(r;t) (3.8)
rB(r;t) = 0; (3.9)
with E, D the eletri eld and the eletri displaement, B the magneti in-
dution and H the magneti eld. and j desribe the external harges and
urrents. The indues ones vanish by the averaging done for this marosopi
approah. This desription is omplete only if the oupling between the D and
E, and between B and H, respetively, is given.
Material oeÆients: To desribe the response linearly,one introduestwo
oupling funtions (also alled onstants frequently), the dieletri funtion "
(also known aspermittivity) and the magneti permeability:
D=""
0
E; B =
0
H; (3.10)
or alternatively dening the eletri polarizability P and the magnetization M
by
D="
0
E+P; P=
p
E="
0
E; (3.11)
H= 1
0
B M; M= 1
m
0
H; (3.12)
deningtheeletriandmagnetisuseptibilitiesand
m
andthepolarizability
p as
"=1+;
p
="
0
; (3.13)
=1
m
; (3.14)
The magneti suseptibility is not given attention anymore. When ouplingthe
urrent j linearly to the eletri eld aording to Ohm'slaw, youintrodue the
eletrialondutivity :
j=E:~ (3.15)
Absorption ofwaves: In vauum,the eletrieldof afreeeletro-magneti
wave follows the wave equation
4E(r;t)=
0
"
0
"
2
2
E(r;t); (3.16)
whih has solutions
E(r;t)=E
0 e
i(kr !t)
: (3.17)
When penetrating matter,the amplitude lowers exponentially,
E =E
0
e ; (3.18)
with the absorption oeÆient dened as
dI
dz
= I (3.19)
forpenetration inz-diretion, and I =jEj 2
the amplitude.
The rest...
Elementary lassis (see Madelung [Mad78℄): We assume a lassial eletro-
magneti wave of form
~
A(~r;t)=
~
A
0 e
i(
~
K~r !t)
=A
0
~ e e
i(
~ n
~
! ~r !t)
withomplexrefrativeindex n(!)~ =n(!)+i(!). Ifrefrationanddieletriity
are oupled by (!) = n~ 2
(!) with (!) =
1
(!)+i
2
(!) omplex, we get the
onnetion
n 2
2
=
1
(3.20)
2n =
2
: (3.21)
Another well mesurable quantity is the relexion of aperpendiular inoming
waveR (!) whihis related ton(!) and k(!)by
R =
1 n~
1+~n
=
(n 1) 2
+k 2
(n+1) 2
+k 2
:
Tensor properties: In the general ase, the oupling (3.10) is not simple
salar, but tensor-like, as well as depending on the frequeny and the loation.
Sinetheouplingishomogeneousintime,andforthemarosopiapproahalso
in spae, the arguments of the suseptibilities read (with the array boundaries
making the statementsausal, oran appropiatedened suseptiblilty)
P(r;t) = Z
d 3
r 0
Z
dt 0
~ (r r
0
;t t 0
)E(r 0
;t 0
) (3.22)
M(r;t) = Z
d 3
r 0
Z
dt 0
~
m (r r
0
;t t 0
)H(r 0
;t 0
) (3.23)
In Fourierspae this onvolution gives
P(k;!) = (k;~ !)E(k;!) (3.24)
M(k;!) = ~ (k;!)H(k;!): (3.25)
Headwords:
reletivity, transmitivity,extintion
Beer's law, sattering ross setion
refrative index, (omplex)dieletri funtion
eps1 - refration,eps2 - absorption
Reetion R=r(n,kappa)
Kramers-Kronig
Fornite temperatures, system is not desribable by awave funtion. statis-
tial averaging needed. (springer,S250)
XXX RPA?
kramers-kronig relations
Transformations.
3.3 Relation between real and imaginary part
Bakground: Dueto the Dira relation
1
!+i
=P 1
!
+iÆ() (3.26)
a spetral distributionfuntion with an energylike parameter !
G(!)=lim
"!0 1
N Z
d 3
k
F(k)
E E(k) i"
(3.27)
has itsreal and imaginary parts
<G(!)=P 1
N Z
d 3
k
F(k)
! !(k)
(3.28)
and
=G(!)=
N Z
d 3
k F(k)Æ(! !(k)); (3.29)
Consequene: Kramers-Kronig Relation These relationsonnet the real
andimaginarypart ofanyparameterthatrelatestwoeldsinalinearandausal
way. The relationsread
1
(!) = 1+
1
P
+1
R
1 d
2 ()
!
2
(!) =
1
P
+1
R
1 d
1 () 1
! :
(3.30)
P denotes the prinipal value of the integral. Sine ! > 0 it is desirable to
transform(3.32) tointegralsover the domain(01). We use the relation
"( !)="
1
( !)+
i( !)
!
="
(!) (3.31)
Bymultiplyingboth the numerator and demoninatorof (3.30)with (+!),one
yields
1
(!) = 1+
2
P
1
R
0 d
! 0
2 ()
2
! 2
2
(!) =
2
P
1
R
0 d
1 () 1
2
! 2
:
(3.32)
TheonsequeneoftheKramers-Kronigrelationsisthatonethe imaginarypart
isknown for the whole spetrum,youknow the realpart as well,and vieversa.
It is alsoworth tonotie that these relations are of universal validity sine they
donot imply any knowledgeof the interations inside the solid.
OneanalsoonstrutKramers-Kronigrelationsforotherquantities, likethe
magnitudeand the phase of the omplex reetion oeÆient.
3.4 Ideas
Eetive mass
oszillatorstrength
ondutivity...
3.5 Transmission spetra
bandgap - absorptionedge
photoni range
eletroni/interband absorption
anisotropy
3.6 Classial Models
lassial,semi-lassial, fully quantum mehanialmodel
oszillatormodel
Lorentz(Tau-Lorentz?)
gas: !
C
, plasmafrequeny
3.7 Dieletri funtion
Mirosopi means loal,marosopi means averaged.
Here relation to one-partile image. Missing many-partile eets, exita-
tions)
3.7.1 Mirosopi denition, dieletriity
Mirosopi Maxwell equations are
re=
mi
"
0
; rb =
0 j
mi +
0
"
0
t e
rb=0; re=
t b
with e = e(r;t) the mirosopi eletri eld and b = b(r;t) the mirosopi
magneti indution. Youaquire the marosopi quantities by averaging:
=h
mi
i; j=hj
mi
i; E=hei; B=hbi: (3.33)
Nowadditionally....
D="
0
E+P; H= 1
0
B M (3.34)
Denition of dieletri funtion and inverse:
E(r;t) = "
1
0 Z
d 3
r 0
Z
dt 0
"
1
ma (r r
0
;t t 0
)D(r 0
;t 0
)
e(r;t) = "
1
0 Z
d 3
r 0
Z
dt 0
"
1
mi (r;r
0
;t t 0
)D(r 0
;t 0
)
where "is atensor. Medium ishomogenousfrommarosopipointofview, but
not mirosopi; there onlylattie periodiity. DF and inverse obey the relation
"
1
(r;r 0
;t t 0
)"
mi (r
0
;r 00
;t 0
t 00
)=Æ(r r 0
)Æ(r 0
r 00
)Æ(t t 0
)Æ(t 0
t 00
) (3.35)
3.7.2 Fourier transforms
Now fourier transforms 1
of the marosopi eletrield:
E(q;!)="
1
0
"
1
ma
(q;!)D(q;t) (3.36)
The mirosopione:
"
mi
(r+R;r 0
+R;t t 0
) = "
mi (r;r
0
;t t 0
)
)"
mi (q;q
0
;!) = e
i(qiqR q 0
)R
"
mi (q;q
0
;!)
with R a reiproal lattie vetor. Sine this means " is only non-zero for a
diereneq q 0
equaltoareiproallattievetor,wemakethefollowinghange
innotation:
"
mi (q;q
0
;!)!"
mi
(k+G;k+G 0
;!); (3.37)
whih means
R
d 3
q ! R
BZ d
3
k P
G
;
R
d 3
q R
d 3
q 0
! R
BZ d
3
k P
G;G 0
;
"
1
(k+G;k+G 0
)"(k+G 0
;k+G 00
)=Æ
GG 0
Æ
G 0
G 00
Togetherwith the denition(3.33) this resultsin the following fouriertransform
forthe mirosopidieletri funtion:
X
G Z
d 3
k Z
d! e
i((k+G)r !t)
e(k+G;!)
= Z
d 3
r 0
X
G;G 0
Z
d 3
k Z
d! e
i((k+G)r+(k+G 0
)r 0
!t)
"
mi (q;q
0
;!)
!
X
G 00
Z
d 3
k 00
Z
! 00
e i((k
00
+G 00
)r ! 00
t)
D(q 00
;! 00
)
!
) e(k+G;!)
= X
G 0
"(k+G;k+G 0
) X
G 00
Z
d 3
k 00
D(k 00
+G 00
) Z
d 3
r 0
e i(k+G
0
+k 00
+G 00
)
= X
G 0
"(k+G;k+G 0
) X
G 00
Z
d 3
k 00
D(k 00
+G 00
)Æ(k+G 0
+k 00
+G 00
)
= X
G 0
"(k+G;k+G 0
)D(k+G 0
)
1
In the following the Fourier transforms are written in the form f(r;t) =
1
2 R
d 3
q R
d! exp(i(qr !t))f(q;!)andf(q;!)= R
d 3
r R
dt exp( i(qr !t))f(r;t), sothe
kindoffuntionisidentiablebyitsparameters. Alsoonlypartlyfouriertransformedfuntions
3.7.3 dieletriity $ internal harge density
Next: Averaging...(244-246) With the fourier transforms of two maxwell equa-
tions
iqe(q)= 1
"
0
mi
(q); iqD(q) = ext
(q); (3.38)
we onlude to
ext
(k+G) = X
G 0
(k+G)"(k+G;k+G 0
)
mi
(k+G 0
)(k+G 0
)
(k+G 0
) 2
X
G 0
jk+Gj jk+G 0
j "(k+G;k+G 0
)
ext
(k+G)+ ind
mi
(k+G 0
)
(k+G 0
) 2
with "(k+G;k+G 0
) =u
k+G
"(k+G;k+G 0
)u
k+G
0 the longitudinaldieletri
funtion (u
k
= k
jkj
unit vetor).
Using the fourier transform ext
(q) ="
0 q
2
U ext
(q), we transform to
X
G
"
1
(k+G 00
;k+G)"
0
(k+G) 2
U ext
(k+G)=
X
G;G 0
jk+Gj jk+G 0
j"
1
(k+G 00
;k+G)"(k+G;k+G 0
)
"
0 U
ext
(k+G 0
)+
ind
mi
(k+G)
(k+G 0
) 2
;
whihresults in
ind
mi
(k+G)="
0 X
G 0
"
1
(k+G;k+G 0
) Æ
GG 0
jk+Gj jk+G 0
jU ext
(k+G 0
)
and
"
1
(k+G;k+G 0
)=Æ
GG 0
+
"
1
jk+Gj jk+G 0
j
ind
mi (k+G)
U ext
(k+G)
3.8 Quantum mehanial model
Indiret transitions:
Diret Transitions...
3.9 Comparison
Possible reasons
DFTdoesn't desribeexited states
just quasi-partiles
3.10 Missing
Missing:
loaleld orretions (Fox, 2.2.3)
RPA
"
2
= 4
2
e 2
m 2
! 2
X
i;j
Z
~!=E
j (
~
k) E
i (
~
k) dk
2
(2) 3
hi
~
kjp
jj
~
kihi
~
kjp
jj
~
ki
r
~
k
E
j (
~
k) E
i (
~
k)
f
0 (E
i (
~
k))(1 f
0 (E
j (
~
k)))
(3.39)
"
2
= X
;
"
2 e
e
; (3.40)
Forthe ubi ase:
"
2
= 4
2
e 2
m 2
! 2
X
i;j
Z
~!=Ej(
~
k) Ei(
~
k) dk
2
(2) 3
jhi
~
kjpjj
~
kij 2
r
~
k
E
j (
~
k) E
i (
~
k)
f
0 (E
i (
~
k))(1 f
0 (E
j (
~
k)))
(3.41)
What about prefator? Atomi units, ! 4
"0 .
Denition of JDOS:
J(E)= X
i;j
Z
E=E
j (
~
k) E
i (
~
k) dk
2
(2) 3
1
r
~
k
E
j (
~
k) E
i (
~
k)
(3.42)
3.11 Krasovskii
"="
intra +"
inter
"
1intra
=1
! 2
p
! 2
"
2intra !
2
p
! Æ(!)
h{kjrj{ki= 1
~ E
{ (k)
k
(3.43)
Basis sets
As already mentionedin setion 2.3, the eigenfuntions are usually expanded in
abasis,
hrjiki=
i
(k;r)= 1
p
X
G C
i
k+G
k+G
(r); (4.1)
where is the unit ell volume. The Hamilton and overlap matries H and S
are onstruted fora set of k-points, and the generalized eigenvalue problem
[H (k)
i
S(k)℄
i
(k)=0 (4.2)
is solved, with
i
(k) = (C i
k+G
) the vetor of the C-oeÆients (of eigenvalue i
and vetor k) for allG's. Many questionsof detail, aswell asgeneral properties
of your alulation like auray and omputational eortdepend onthe hoie
of your basis set.
The better the basis funtions math the shape of the atual wavefuntions,
the better the onvergene is. Somebasis sets may have drawbaks that an not
always be liftedby a bigger ut-o.
4.1 The Plane-Wave Basis
Averysimple basisset isbuildout of planewaves (PWs), the eigenfuntionsfor
aonstant potential,that are free eletrons
k+G
(r)=e i(k+G)r
;
The use ofthis basis ompliestoa simplefourier transform. Typiallythis is
agoodhoie for nearly free eletrons and deloalized eletrons.
The simple analyti form usually leads to well-performing alulations that
arestraight-forward toimplement. Thehamiltonand overlapmatriesan easily
be alulated as
H
G;G 0
(k) =
~ 2
2m
jk+Gj 2
Æ
GG 0
+V
(G G 0
)
; (4.3)
V
(G G 0
)
= Z
u d
3
r e
i(G G 0
)r
V
eff (r)
S
G;G 0
= Æ
GG 0
The matrix elements of the momentum operator for instane in this basis (in
terms of the eigenfuntions) give
Figure 4.1: Used G-vetors in expansion. Small x-like rosses indiate the basis
vetors of reiproal spae. The plus-like rosses indiatethe (k+G)-vetors orre-
sponding to the k-vetor drawn in the origin. The large irle enloses all vetors of
jGj<G
max
,thesmallerone those of jk+Gj<G
max .
fkj r
i jik
PW
= 1
X
G
(k+G)C f
k+G C
i
k+G
: (4.4)
The hoieof G-vetors isillustratedingure 4.1. Afterhoosing aut-ovalue
G
max
, all (k +G)-vetors are used that obey jk+Gj G
max
. This hoie is
neessary beauseof numerialreasons. Thenumberof basisfuntionsobviously
Potentials: XX, andless eÆient forsystems inludingloalizedvaleneele-
trons,like transitionmetals.
The prie for this simpliity is the inability of this basis set to desribe the
strong interations inluding the nulear potential 1
r
. As a solution, the idea
of pseudopotentials has been developed. The potentials are idential to the all-
eletron potential outside a given ore-radius, but of dierent, smoother shape
inside. They are onstrutedjustthat the resultingpseudo-wavefuntionmimis
the all-eletron wavefuntion outside this radius as lose as possible. For many
elements,this methodworks reliable, yieldingsmooth potentials.
4.2 The APW method
Figure 4.2: Spatial partitioning in augmented basis sets. The irles are the muÆn
tins,leavingtheinterstitialregion, plottedgrayed.
A basis set of better shape has been proposed by Slater already in 1937
[Sla37℄. InthisAugmentedPlaneWave (APW)basis,spaeisdividedintospheres
that are entered around eah atom, so-alled muÆn-tins (MTs), and into the
remaininginterstitialregion (IS) 1
.Whileplanewaves are usedas basisfuntions
1
intheinterstitial,they areaugmented inthespheres byspherialharmonistime
radial basis funtions that are solutionsto of the radial Shrodingerequation to
anl-dependent energy
~ 2
2m
2
r 2
+
~ 2
2m
l(l+1)
r 2
+V(r) E
l
ru
l
(r)=0: (4.5)
Expanding the funtion in a series of these funtionsup to anl-uto l
max , this
givesthe basis funtions(the augmented planewaves)
k+G (r)=
8
>
<
>
: e
i(k+G)r
r2IS
l
max
X
l =0 l
X
m= l a
l m
(k+G)u
l (r;E
l )Y
l m
(^r) r2MT
:
(4.6)
The alulation of matrix elements beomes more ompliated due tothe radial
funtionsbeingnon-orthogonalwhenrestritedtothemuÆn-tins,andduetothe
omplex shapeof the interstitialregion.
It isuseful tonormalizethe radial funtions like
hu
l ju
l i=
R
Z
0
dr ju
l j
2
=1 (4.7)
Toensurethatthesebasis funtionsare ontinuous, onehastomaththe muÆn-
tinfuntions tothe planewavesonthe boundaries. Toarrangethis, one expands
the spherial harmonis intoplanewaves using the Rayleigh relation
e iKr
=4 X
l m i
l
j
l (rK)Y
l m (
^
K)Y
l m
(^r): (4.8)
K = jKj is the length of the vetor K = k+G, and j
l
is the Bessel funtion
of the rst kind. An atom at position S
owns a oordinate frame (U
;S
)
(in the style of symmetry operations ??, U
being the rotation matrix). In this
frame, a plane-wave takes the form
e iKr
!e i(U
K)(r+U
S
)
(4.9)
Mathing the planewavesonthe sphereboundarieswith the muÆn-tinfuntions
for every augmented wave gives the a-oeÆients as
a
l m
(K)=e iKS
4i l
u
l (R
;E
l )
j
l (KR
)Y
l m (U
^
K): (4.10)
This leaves theC-oeÆients(and the energiesE
l
)asthe variationalparameters
of the method, the a's being determined by them. In fat this mathing works
only on a few points exatly, but the so-hosen A-oeÆients yield the smallest
mismath.
Withthese basis funtions the wavefuntion take the form
i
(k;r)= 8
>
>
<
>
>
: 1
p
X
G C
i
k+G e
i(k+G)r
r2IS
X
G X
l m C
i
k+G a
l m
(k+G)u
l (r;E
l )Y
l m
(^r) r2MT
(4.11)
Sinethea-oeÆientsare{togetherwiththeexpansionoeÆientsC{theonly
terms insidethe spheres depending onG, one an write the whole wavefuntion
shorteras
i
(k;r)= 8
>
<
>
: 1
p
C
i
k+G e
i(k+G)r
r2IS
X
l m A
i;
l m (k)u
l (r;E
l )Y
l m
(^r) r2MT
(4.12)
with the shorthand
A i;
l m (k) =
X
G C
i
k+G a
l m
(k+G): (4.13)
Potentials: Sinethesebasisfuntionsarethesolutionsofaonstantpotential
in the interstitial and a spherial potential in the muÆn tins, this muÆn-tin
approximation for the shape of the potentials has initially been used. In the
warped muÆn-tinapproximation,the interstitialpotentialisextended togeneral
shape, that means extended in planewaves.
Problems of the method: Aording to (4.10) the A's are determined om-
pletely by the planewave oeÆients. So these C oeÆients together with the
energy parameters E
l
are the variational parameters of this method. If the en-
ergyparameters were taken as xed rather than as a variational parameter, the
method would simply onsist of the use of the APW basis set with solving the
seularequation (4.1). The solutionswould givethe band energies.
Unfortunately, this is not a workable sheme. The basis funtions lak varia-
tionalfreedom,this meansthey donot yieldorretresultsiftheenergy parame-
tersE
l
mismaththeatualbandenergies. Thismeansthattheseenergiesforone
k-pointan notbeobtained fromasinglediagonalization,butithastobesolved
iteratively. This makesthis methodmuh more omputationallydemanding.
FurthermoreitisdiÆulttouseageneralpotentialbeyondthewarpedmuÆn-
tinapproximation [Sin94℄. Another obstaleisthe so-alled asymptoteproblem.
Theremightbeenergyparametersforwhihu
l
vanishesorbeomesverysmallon
the sphere boundary. As aonsequene the planewaves and the radial funtions
4.3 The LAPW method
Muh work has been devoted tolifts the desribed problems. In 1975, Andersen
introduedtheLinearizedAugmentedPlaneWave (LAPW)method. Theentral
idea istodesribethe basisfuntionsinsidethe muÆn-tins not onlybysolutions
of the radial Shrodinger equation u
l (r;E
l
), but as well by its energy derivates
_ u
l (r;E
l )
E u
l (r;E
l ). IfE
l
diersslightlyfromthetruebandenergy,aording
to anexpansion with respet tothe energy,
u
l
(r;)=u
l (r;E
l
)+( E
l )u_
l (r;E
l
)+O(( E
l )
2
); (4.14)
the true radial funtionan be approximated suÆiently. The error inthe band
energies will be of the order O(( E
l )
4
). The energy derivatives an aquired
from (4.5),taking the energy derivative:
~ 2
2m
2
r 2
+
~ 2
2m
l(l+1)
r 2
+V(r) E
l
ru_
l
(r)=ru
l
(r): (4.15)
The basis funtions are now
k+G (r)=
8
>
>
>
>
>
<
>
>
>
>
>
: 1
p
e
i(k+G)r
r 2IS
l
max
X
l =0 l
X
m= l [a
l m
(k+G)u
l (r;E
l )+
b
l m
(k+G)u_
l (r;E
l )℄Y
l m
(^r) r 2MT
:
(4.16)
Analogous tothe APW method,the muÆn-tinoeÆients are determined as
a
l m
(K) = e iKS
4i
l
W Y
l m (U
^
K)
[u_
l (R
)Kj
0
l (KR
) u_
0
l (R
)j
l (KR
)℄ (4.17)
b
l m
(K) = e iKS
4i
l
W Y
l m (U
^
K)
[u
l (R
)Kj
0
l (KR
) u
0
l (R
)j
l (KR
)℄ (4.18)
with the Wroskian
W =[u_
l (R
)u
0
l (R
) u
l (R
)u_
0
l (R
)℄: (4.19)
Colleting terms equivalentto the APW basis set, withthe denitions
A i;
l m
(k) = X
G C
i
k+G a
l m
(k+G);
B i;
l m
(k) = X
C i
k+G b
l m
(k+G) (4.20)
the wavefuntions take the form
i
(k;r)= 8
>
>
<
>
>
: 1
p
X
G C
i
k+G e
i(k+G)r
r2IS
X
l m
A i;
l m (k)u
l (r;E
l )+B
i;
l m (k)u_
l (r;E
l )
Y
l m (^
r) r2MT
(4.21)
The detailedonstrution of the H and S matriesis desribed in[Kur00℄.
Withthisadditionalexibility,theLAPWsformagoodbasisformostsetups.
Inontrasttothe APWmethodonlyonediagonalisationisneeded toobtainthe
band energies. And sine it is very unlikely that both radial funtion and its
derivativevanish the asymptote problemdoes not our.
Basisonversion: Amethodtolinkthesimpliityoftheplanewavebasiswith
the auray of the more sophistiated LAPW basis set has been proposed by
Krasovskii [KSS99℄. In this Augmented Fourier omponents method (AFC), the
viinityoftheoreontainingrapidalterationsofthewavefuntionsisonsidered
to be of low inuene on the hemial behaviour. The results of alulation
in LAPW basis are therefore gauged by an appropiate funtion, generating a
smootherwavefuntioninthisregionandleavingaslowlyvaryingvaleneharge,
whih an be represented adequately in a planewave basis. From this results,
quantities an be alulated inthe simple planewave formalism.
4.4 The Loal Orbital extension
Theremight be situationswhere the variationalfreedom of the LAPW basis set
isnotsuÆient. Oneexamplearesemi-orestates,whiharestatesoflowenergy
thatdonot deay ompletelywithinthe muÆn-tins,but haveanoverlap intothe
interstitial. Singh [Sin91℄ introdued the Loal Orbital extension to the LAPW
set to dealwith suh problems.
The idea is to expand the basis set by additional funtions that are zero in
the interstitial, to extend exibility inside the muÆn-tins. By onstruing these
additionalbasisfuntionssuhthat thederivativevanishesonthe spherebound-
ariesas well,the A- and B- oeÆientsonstruted in(4.17)remain unhanged.
The new basis funtion should have the harateristi of a ertain angular mo-
mentum l
l o
and energy E
l o
. This is ensured by a ombination of three radial
wavefuntions,
l o [a
l o u
l (r;E
l )+b
l o _ u
l (r;E
l )+
l o u
l (r;E
l o
)℄: (4.22)
Here the index lo =1;:::;n
l o
runs over the numberof loalorbitalsintrodued,
the a
l o
;b
l o
;
l o
are the orresponding oeÆients for eah atom. The l = l
l o in-
diates the angular momentum quantum number assoiated with this loal or-
(l(lo);E
l o
), enrihed with the LAPW-likerst two parts ensuring the onditions
of the boundary.
These two onditions together with the normalization ondition determine
the a;b; oeÆients of eah loal orbital (for details on this, as well as on the
onstrution of the matrix elements, see [Kur00℄). The basis funtions have to
sitisfy Bloh's theorem. They are therefore mathed to titious planewaves to
obtain the properXXX
4.5 Notes on symmetry
symmetries an be used tosimplify the alulations.
... (aswell asthe other quantities like harge density and potentials)
point group symmetry and spae groupsymmetry.
inversion symmetry: real and omplex
Creal! (4.23)
equivalent atoms
4.6 Notes on the kineti energy operator
Implementation
The implementationof the dieletri funtion for this thesis has been done with
the FLEUR ode [FLE℄ in bulk mode. FLEUR is a full-potential linear aug-
mented plane-wave(FLAPW)ode. Inthe followingsetionssomedetailsofthis
implementationshall be disussed.
Sine the linearized augmented plane-waves are the basis of hoie, the for-
mulae of the momentum matrix elements in this basis are presented in setion
5.1. Details onperformingthe k-spae integration to obtain the real part of the
dieletri funtion are shown in 5.2. The real part is obtained in 5.3. Due to
a restrition of FLEUR, the eet of bakfolding has to be disussed in setion
5.4. The problemati inuene of this bakfolding on the numerial integration
isdesribed insetion 5.5. Someremarks in5.6are followed by a twotest alu-
lations.
Whenreferringinthefollowingtothedieletrifuntion,oftenit'simaginary
part ismeant. This should be lear fromthe ontext.
5.1 Momentum matrix elements in the LAPW
basis
The momentum matrix elements (MMEs) 1
~
i
hfkjrj{ki=
~
i Z
u d
3
r
f
(k;r)r
{
(k;r)M
fi
(k) (5.1)
are tobealulated inthe LAPW basis. Dueto the partitioning ofthe unit ell
intomuÆn-tins and the interstitial regionby the hoie of the augmented basis,
1
Toavoidonfusionwiththeimaginaryuniti,theinitialeletronitransitionlevelislabeled
{.
the matrix elementshave tobe alulated inthese regions separately:
hri=hri
IS +
X
hri
MT
: (5.2)
The formulaeare presented inatomiunits(seeappendix C), sothe fator ~=1
disappears.
5.1.1 Interstitial ontribution
In the interstitial,the wavefuntions are 2
j{ki
IS
= 1
X
G C
{
k+G e
i(k+G)r
; r2IS; (5.3)
and the nablaoperator ats like
rj{ki= 1
X
G
i(k+G)C {
k+G e
i(k+G)r
; r2IS; (5.4)
so that the interstitial partof the matrix element reads
fkj r
i j{k
IS
= 1
X
GG 0
(k+G)C f
k+G 0C
{
k+G Z
IS d
3
r e i(G G
0
)r
: (5.5)
The non-trivialinterstitialvolumethe integralats onis handled by subtrating
the muÆn-tins fromthe whole unit ell :
Z
IS d
3
r e i(G G
0
)r
= Z
d
3
r e i(G G
0
)r X
Z
MT
d
3
r e i(G G
0
)r
: (5.6)
WhiletherstintegralgivesthesimplevalueÆ
GG 0
,theintegraloveramuÆn-tin
entered atS
givesthe split solution
Z
MT
d
3
r e i(G G
0
)r
= (
V G=G
0
3V
sinx xosx
x 3
e i(G G
0
)S
G6=G 0
(5.7)
with x = jG G 0
jR
and R
;V
the radius and the volume, respetively, of
sphere . Altogetherthis gives
fkj r
i j{k
IS
= 1
X
G
(k+G)
"
C {
k+G
X
V
!
X
G 0
6=G C
f
k+G 0
X
3V
sinx xosx
x 3
e i(G G
0
)S
#
(5.8)
= X
GG 0
(k+G)C {
k+G C
f
k+G 0
s(G G 0
): (5.9)
2
In the last line, the expressions of the preeeding integral were merged into the
funtions, that is
s(G G 0
)= 8
<
: 1
(
P
V
) G=G
0
3
P
V
sinx xosx
x 3
e i(G G
0
)S
G6=G 0
(5.10)
with the above x =jG G 0
jR
. This is the Fourier representation of the step-
funtion
S(r)= (
1; r 2IS
0; r 2MT;
(5.11)
whih is usually onstruted already for the onstrution of the Hamilton and
overlap matriesH and S in the self-onsisteny part.
5.1.2 MuÆn-tin ontributions
The further proedure depends on what form of wavefuntions you start from.
If you use the LAPW funtions written expliitly in the basis funtions (4.16),
withoutthe summation(4.20)inthe alulationof your MMEs(5.1), youobtain
the summations over G,l,m eah twie. In the further derivation, not only one
pair ofthe (l;m)-summationvanishes,but also, by leveronversion, the seond
m-summation [Kra℄. This leaves summations G;G 0
;l. If you do this, you an
simplyhek the hermitiity of your matrix for every G-vetor.
In the derivation used inthis thesis, LAPWs ofthe aumulated form(4.21)
are used. Toderive the matrix elementsin the spheres, the momentum operator
is expressed in spherialoordinates, and its impat on the spherial harmonis
isalulated. Sine this part is abit lengthy, itis moved toappendix A.
InallusiontotheladderoperatorsL
+
andL oftheangularmomentumoper-
ator,oneexpresses the momentummatrixelementsnotintermsof(x;y;z) T
,
but inthe rotatedform
0
B
x+iy
x iy
z 1
C
A
=M 0
B
x
y
z 1
C
A
0
B
1
2
3 1
C
A
; (5.12)
with the base hange matrix and its inverse
M= 0
B
1 i 0
1 i 0 1
C
A
; M
1
= 0
B
1
2 1
2 0
1
2 i
1
2 i 0
1
C
A
: (5.13)
The result ontains only one (l;m)-summation an be expressed as
hfkj
n j{ki =
lmax 1
X
l =0 l
X
m= l
(5.14)
[( R
u
l +1 u
0
l r
2
dr l
R
u
l +1 u
l
rdr) A f
l +1;m 0
A {
l ;m
+ (
R
u
l +1 _ u 0
l r
2
dr l
R
u
l +1 _ u
l
rdr) A f
l +1;m 0
B {
l ;m
+ (
R
_ u
l +1 u
0
l r
2
dr l
R
_ u
l +1 u
l
rdr) B f
l +1;m 0
A {
l ;m
+ (
R
_ u
l +1 _ u 0
l r
2
dr l
R
_ u
l +1 _ u
l
rdr) B f
l +1;m 0
B {
l ;m
℄ F (2n 1)
l ;m
+ [( R
u
l u
0
l +1 r
2
dr + (l+2) R
u
l +1 u
l
rdr) A f
l ;m A
{
l +1;m 00
+ (
R
u
l _ u 0
l +1 r
2
dr + (l+2) R
u
l +1 _ u
l
rdr) A f
l ;m B
{
l +1;m 00
+ (
R
_ u
l u
0
l +1 r
2
dr + (l+2) R
_ u
l +1 u
l
rdr) B f
l ;m A
{
l +1;m 00
+ (
R
_ u
l _ u 0
l +1 r
2
dr + (l+2) R
_ u
l +1 _ u
l
rdr) B f
l ;m B
{
l +1;m 00
℄ F (2n)
l +1;m 00
for n=1;2;3indiatingthe omponents, and m 0
,m 00
given by
m 0
= 0
B
m+1
m 1
m 1
C
A
; m 00
= 0
B
m 1
m+1
m 1
C
A
for n = 0
B
1
2
3 1
C
A
: (5.15)
The fators F (n)
l m
are dened in appendix A. In the ombinations of oeÆients
owning angularquantum numbersl and l+1inthe produts,one reognizesthe
dipoleseletion rules, i.e.the onservation of angular momentum.
Thenotationalreadyindiatesthatonlythelargeomponentofthewavefun-
tioninsidethemuÆn-tinsistaken intoaount. Forthevalenestatesonsidered
this is a good approximation. The ontributions resultingfrom the loalorbital
extensiontothe LAPW basis set (??)are similarinshapetothose ofthe simple
LAPWbasis(5.14),butmorelengthy,andarehenegiveninappendixAaswell.
5.1.3 Properties of the matrix elements
Hermitiity: Sine the momentum operator is an observable and therewith
hermitian, somust beitsmatrix elements. Thisan be shown easilyby applying
partialintegration tothe deningformulaof thematrix elements(5.1). Itisalso
obvious for the MMEs writtenin the plane-wave basis (4.4).
However it an be hardly seen from the formulae written in LAPW basis,
sine the interstitial plane-waves are expanded on the muÆn-tin boundaries in
terms of spherialharmonis utilizing the Rayleigh relation(4.8). If one applies
partial integration to the LAPW formulae, one an see that e.g. for the (x+
iy)-omponent of the muÆn-tin ontribution to the MME, parts of the fators
ontainingF (1)
l ;m
ompensatewith theomplexonjugateofthe fatorsontaining
F (2)
,leaving the boundary values of the integration un-ompensated.
The rest has tobetaken by the dierenein onjugating the interstitialon-
tribution,whih issensitiveto onjugation due tothe fator (k+G)in the rst
sum in(5.9).
Reality: The diagonalmatrix elementsare real sine the momentum operator
isanobservable. Furthermorethis an alsobe seenfrom and omparedwith the
derivativesoftheenergybands(3.43). Thenon-diagonalpartsareingeneralom-
plex,as an be assumed beause of the omplex A,B muÆn-tinoeÆients. For
the ase of inversion symmetry, however, the matrix elementsbeome real. This
isobviousforthe plane-wavebasis(4.4)duetothenowrealC oeÆients (4.23),
but not forLAPW basis (due tothe re-expansion onthe muÆn-tinboundaries).
Equivalent atoms: XXXX
Shouldbereal fordiagonal
In generalomplex, but "
2
is real again!
'magi ofnumbers'
5.1.4 Illustration
0.001 0.01 0.1 1
| M fi (k) |
1 - 2 1 - 3 1 - 4 2 - 3 2 - 4
0.0 0.0 0.0
0.0 0.5 0.0
0.5 0.5 0.0
0.5 0.5 0.5
0.0 0.0 0.0
Figure5.1: Theabsolutevalueofmatrixelementsforveseletedtransitionsevolving
on paths on the border of the irreduible Brillouinzone. The initial and nal level {
andf aregiven inthelegend.
To give animpression of the amplitude and k-dependene, a band struture-
