QUANTUM ELECTRONIC PROPERTIES OF
LOW-DIMENSIONAL TRANSITION-METAL
SYSTEMS: MODELING, SIMULATION AND
CHARACTERIZATION.
Thesis by
i ián sánchez amí ez
o he deg ee o
doc o o philosophy in physics
Supe ised by
e nando de juan sanz
and
maia ga cía e gnio y
Uni e si y o he Basque Coun y (EHU/UPV)
Donos ia, May 2025.
(cc) 2025 I ián Sánchez Ramí ezs (cc by-nc-sa 4.0)
I ián Sánchez Ramí ez: Quan um elec onic p ope ies o low-dimensional ansi ion-
me al sys ems: modeling, simula ion and cha ac e iza ion, May 2025.
A quien me enseñó a anda y aún sigue acompañándome en cada paso.
Sob e odo, no pie das u deseo de camina [...] la salud y la sal ación sólo pueden
se encon adas en el mo imien o.
— Sø en Kie kegaa d
CONTENTS
Ac onyms iii
Publica ions x
Abs ac xii
Resumen xi
i in oduc ion 1
1 in oduc ion 2
1.1His o ical and concep ual backg ound o his wo k . . . . . . 2
1.1.1Fi s classical s eps. . . . . . . . . . . . . . . . . . . . . . 3
1.1.2Ad en o quan um mechanics. . . . . . . . . . . . . . . 4
1.2S a e-o - he a , goals and ou line o his wo k . . . . . . . . . 8
1.3A b ie commen on ou app oach . . . . . . . . . . . . . . . . 11
ii heo y &me hods 15
2 un a eling he many-body elec onic hamil onian 16
2.1A Hamil onian o gobe n hem all . . . . . . . . . . . . . . . . 17
2.2App oxima ions........................... 19
2.2.1Bo n-Oppenheime app oxima ion . . . . . . . . . . . . 20
2.2.2Ha ee app oxima ion . . . . . . . . . . . . . . . . . . . 23
2.2.3Ha ee-Fock app oxima ion . . . . . . . . . . . . . . . 26
2.2.4Koopman’s heo em: om a compu a ional simpli i-
ca ion o a physical insigh . . . . . . . . . . . . . . . . 28
2.3Densi y Func ional Theo y . . . . . . . . . . . . . . . . . . . . . 29
2.3.1The Hohnenbe g-Kohn heo ems . . . . . . . . . . . . . 30
2.3.2The Kohn-Sham app oach . . . . . . . . . . . . . . . . . 33
2.3.3Local Densi y App oxima ion (LDA) . . . . . . . . . . . 35
2.3.4Gene alized G adien App oxima ion (GGA) . . . . . . 36
2.3.5A b ie no e on modi ied Becke-Johnson exchange po-
en ial............................. 39
2.3.6Concluding ema k..................... 40
2.4Pseudopo en ials .......................... 41
2.4.1Ea ly p ecu so s o pseudopo en ials. . . . . . . . . . . 42
2.4.2No m-conse ing pseudopo en ials . . . . . . . . . . . 42
2.4.3P ojec o augmen ed wa e me hod . . . . . . . . . . . . 44
2.4.4Conclusion.......................... 46
2.5Bloch heo em and plane-wa ebasis . . . . . . . . . . . . . . . 46
2.6How DFT educes compu a ional cos . . . . . . . . . . . . . . 49
2.7Wannie Func ions ......................... 50
2.7.1Gene alized Wannie Func ions o Mul iple Bands . . 51
i
con en s
2.7.2Non-Uniqueness and Maximally localized Wannie unc-
ions ............................. 52
2.7.3Applica ions and Wannie in e pola ion . . . . . . . . . 52
2.7.4Why and when using Wannie in e pola ion . . . . . . 53
iii esul s 56
3 quasi-1d mcs 57
3.1In oduc ion ............................. 57
3.1.1Backg ound ......................... 57
3.1.2Open ques ions and ou line . . . . . . . . . . . . . . . . 58
3.1.3Ab ie no eonCDW.................... 60
3.1.4TMCs (MSe4)nIwi h M=Nb,Ta............. 63
3.1.5Phenomenology o e iew on (MSe4)3I(M=Nb,Ta) TMCs. 64
3.2Me hods & Theo e ical esul s o (MSe4)3ITMCs. . . . . . . 66
3.2.1DFT de ails and me hodology . . . . . . . . . . . . . . . 66
3.2.2DFT band s uc u es. . . . . . . . . . . . . . . . . . . . . 67
3.2.3Tigh -Binding models . . . . . . . . . . . . . . . . . . . 70
3.2.4Un olding ab-ini io and ARPES . . . . . . . . . . . . . . 74
3.2.5Op ical conduc i i y . . . . . . . . . . . . . . . . . . . . 76
3.2.6Doping holes s. elec ons . . . . . . . . . . . . . . . . . 78
3.2.7Concluding ema ks on he heo e ical esul s. . . . . . 80
3.3Expe imen al ealiza ion: (NbSe4)3I. ............... 81
3.3.1In oduc ion: Wha a e da k s a es. . . . . . . . . . . . . 82
3.3.2De ailed s uc u al classi ica ion. . . . . . . . . . . . . . 84
3.3.3App oxima ely da k s a es as seen in ARPES. . . . . . 84
3.3.4Discussion & conclusion on he expe imen al esul s. . 87
3.4Disco e y o a new quasi-1D TMC: (Nb4Se15I2)I2........ 88
3.4.1In oduc ion & b ie heo e ical and expe imen al de-
ails............................... 89
3.4.2Chemical & s uc u al de ails. . . . . . . . . . . . . . . . 89
3.4.3DFT & heo e ical analysis. . . . . . . . . . . . . . . . . 91
3.4.4ConcludingRema ks.................... 93
3.5Concluding ema ks and ou look . . . . . . . . . . . . . . . . . 94
4 2d,bulk mds & hei he e os uc u es 97
4.1In oduc ion: Ge o know TMDs . . . . . . . . . . . . . . . . . 97
4.1.1Thei s uc u e........................ 98
4.1.2Thei CDWs.......................... 99
4.1.3Thei supe conduc i i y. . . . . . . . . . . . . . . . . . . 100
4.1.4Open ques ions and ou line. . . . . . . . . . . . . . . . . 101
4.2Backg ound o ou heo e ial esea ch . . . . . . . . . . . . . . 102
4.2.1A b ie no e on Mo & Kondo physics. . . . . . . . . . 104
4.3Cha ge ans e in T/H he e os uc u es o monolaye TMDs. 111
4.3.1Ab-ini io me hods......................114
4.3.2Cha ge densi y wa e dis o ions . . . . . . . . . . . . . 115
con en s i
4.3.3Wo k unc ion analysis . . . . . . . . . . . . . . . . . . . 116
4.3.4Cha ge ans e .......................116
4.3.5Vande Waalse ec ....................119
4.3.6Udependence........................120
4.3.7Dis ance dependence . . . . . . . . . . . . . . . . . . . . 122
4.3.8Resul s: Bulk 4Hbpoly ypes ...............123
4.3.9Conclusion..........................124
4.4Expe imen al ealiza ion: Me allici y h ough andom s acking.127
4.4.1In oduc ion & Backg ound . . . . . . . . . . . . . . . . 127
4.4.2dI/dV spec oscopy.....................128
4.4.3Why s acking diso de ? . . . . . . . . . . . . . . . . . . 129
4.4.4Why no domain walls? . . . . . . . . . . . . . . . . . . 131
4.4.5DFTs udy..........................132
4.4.6Concluding ema ks o his Sec ion . . . . . . . . . . . 134
4.5Concluding ema ks ........................135
i concluding ema ks 137
5 concluding ema ks 138
appendix 144
a in oduc ion 145
a.1B ie philosophical commen . . . . . . . . . . . . . . . . . . . 145
a.1.1Wha is ou app oach? . . . . . . . . . . . . . . . . . . . 146
a.1.2Wha is ue, wha is eal? . . . . . . . . . . . . . . . . . 149
a.1.3A small no e on closu e and ca ego ies . . . . . . . . . 152
a.1.4On disco e y and eme gence . . . . . . . . . . . . . . . 153
a.1.5Final ema k.........................154
b heo y &me hods 155
b.1Va ia ional heo em.........................155
b.2Func ional de i a i es . . . . . . . . . . . . . . . . . . . . . . . 156
c esul s 158
c.1Un olding ..............................158
c.1.1Wha isasupe cell? ....................158
c.1.2Folding and un olding wa e ec o s . . . . . . . . . . . 159
c.1.3Folding and un olding o s a es . . . . . . . . . . . . . . 160
c.1.4Spec al weigh in plane wa e basis . . . . . . . . . . . 161
c.2A b ie no e on Fe mi su ace nes ing & CDWs. . . . . . . . . 162
c.2.1CDW Fo ma ion ia Nes ing . . . . . . . . . . . . . . . 164
c.2.2Geome ical In e p e a ion o Nes ing . . . . . . . . . . 164
c.2.3Quan i ying Nes ing: The Lindha d Suscep ibili y . . . 165
c.2.4Expe imen al Signa u es o Nes ing-D i en CDWs . . 165
c.3Angle-Resol ed Pho oemission Spec oscopy (ARPES) . . . . 165
c.3.1Pho oemission and ene gy conse a ion . . . . . . . . . 166
c.3.2Momen um-Resol ed Elec onic Band S uc u e . . . . 167
con en s ii
c.3.3ARPES in he S udy o CDW Ma e ials . . . . . . . . . 167
c.3.4Sel -Ene gy E ec s and Many-Body In e ac ions . . . . 168
c.3.5Supe conduc i i y and ARPES . . . . . . . . . . . . . . 168
c.4Suppo ing expe imen al de ails o Sec. 3.3............169
c.5Phase ansi ion de e mina ion, suppo ing expe imen al da a
o Sec. 3.4...............................170
c.6Displacemen s o 2D-TMDs s udied. . . . . . . . . . . . . . . . 171
c.7U bands o 2D-TMDss udied....................171
c.8A b ie no e on STS & dI/dV....................171
c.9BCS Theo y o Supe conduc i i y . . . . . . . . . . . . . . . . . 175
c.9.1Elec on-Phonon In e ac ion and Coope Pai Fo ma ion176
c.9.2BCS G ound S a e and Ene gy Gap . . . . . . . . . . . 177
c.9.3Implica ions o ou esea ch . . . . . . . . . . . . . . . 179
c.10 Suppo ing in o ma ion o Sec. 4.4.................180
c.10.1ACBands ..........................180
c.11 Suppo ing expe imen al in o ma ion. . . . . . . . . . . . . . . 180
c.11.1Expe imen al supe conduc i i y cha ac e iza ion . . . 181
c.11.2Concluding ema ks on he expe imen al wo k . . . . 185
i bibliog aphy 186
bibliog aphy 187
ACRONYMS
•1D: One Dimensional
•2D: Two Dimensional
•ARPES: Angle-Resol ed Pho o-Emission Spec oscopy
•BCS: Ba deen-Coope -Sch ie e (model o supe conduc i i y)
•BO: Bo n Oppenheime (App oxima ion)
•CDW: Cha ge Densi y Wa es
•CVT: Chemical Vapou T anspo
•DFT: Densi y Func ional Theo y
•DOS: Densi y o S a es
•GEA: G adien Expansion App oxima ion
•GGA: Gene al G adien App oxima ion
•IBZ: Fi s B illouin Zone
•LDA: Local Densi y App oxima ion
•MLWF: Maximally Localized Wannie Func ions
•mBJ: Modi ied Becke-Johnson (exchange po en ial)
•NCPP: No m-Conse ing PseudoPo en ials
•OPW: O hogonalized Plane Wa es
•PAW: P ojec ed Augmen ed Wa e
•PBE: Pe dew Bu ke E ze ho
•SG: Space G oup
•STM: Scanning Tunnel Mic oscope
•SOC: Spin-O bi Coupling
•TCC: Ca ego ial Closu e Theo y, Teo ía del Cie e Ca ego ial
•TMC: T ansi ion Me al Chalcogenides
iii
ac onyms ix
•TMD: T ansi ion Me al Dichalcohenides
•PW91: Pe dew Wang 1991
•XRD: X-Ray Di ac ion
esumen x i
Es as hipó esis ue on pos e io men e e i icadas expe imen almen e g a-
cias a la colabo ación con el g upo de N. Sch ö e , quien lle ó a cabo me-
diciones de ARPES en (NbSe4)3I, con i mando la na u aleza indi ec a del
bandgap an icipada eó icamen e. Pos e io men e, se aplicó el mismo ma co
eó ico a la ca ac e ización de un nue o compues o sin e izado po el g u-
po de D. Shoemake , (Nb4Se15I2)I2. A pa i de cálculos de DFT, medidas
de anspo e y análisis de es ados de oxidación, se con i mó su compo -
amien o aislan e, demos ando an o la aplicabilidad gene al del en oque
desa ollado como la ubicuidad de los TMCs cuasi-unidimensionales. Ade-
más, se p opuso un lujo de abajo o ien ado a la iden i icación de nue os
sis emas cuasi-1D con "es ados ap oximadamen e oscu os" y p opiedades
op oelec ónicas ajus ables.
El Capí ulo 4se dedica al es udio de los TMDs bidimensionales y sus he e-
oes uc u as. Es os ma e iales han cob ado un p o agonismo c ecien e en
los úl imos años debido a la ácil modulación de sus capacidades elec óni-
cas , óp icas y magné icas, así como po su ap i ud pa a albe ga enómenos
ue emen e co elacionados como CDWs, supe conduc i idad, es ados de
Mo y el e ec o Kondo. En es e capí ulo se analizan bicapas o madas po
capas al e nan es de ipo T(oc aéd icas) y H(p ismá icas igonales), así
como o as ases como las 4Hby6R.
Con el obje i o de elabo a un es udio sis emá ico de es os sis emas, en
es e Capí ulo se desa olla una me odología basada en DFT pa a calcula
la ans e encia de ca ga en e capas, un pa áme o c ucial pa a la desc ip-
ción de enómenos de co elación elec ónica como el aislamien o ipo Mo
o el e ec o Kondo. En es e es udio se con empla on a iaciones en la dis an-
cia in e capas, la composición de calcógenos, y la inclusión de co ecciones
elec ón-elec ón median e el pa áme o Ude Hubba d y co ecciones de
Van de Waals. Los esul ados p esen ados e elan que la dis ancia en e
capas es el p incipal ac o que egula la ans e encia de ca ga, mien as
que las co ecciones asociadas a Uy a las in e acciones de Van de Waals
p esen an un e ec o secunda io. Asimismo, se iden i ican endencias sis e-
má icas según la composición química del sis ema (M = Nb, Ta; X = Se,
S), obse ándose que los compues os o mados po Ta y Se exhiben una
ans e encia de ca ga mayo en compa ación a los compues os o mados
po Nb y S. Es os esul ados pe mi en con ex ualiza las obse aciones ex-
pe imen ales en ma e iales como TaS2, TaSe2y NbSe2.
En colabo ación con el g upo de M. Ugeda, en es e Capí ulo se ca ac e-
iza el compues o 1T-TaSSe, un análogo me álico isoes uc u al al aislan e
1T-TaSe2. Los cálculos ealizados mues an que el deso den en el apila-
mien o de las CDWs acili a el unelado in e capas, p opo cionando una
explicación pa a la me alicidad de es e compues o. La compa ación con
esumen x ii
esul ados expe imen ales ob enidos median e écnicas de mic oscopía y
espec oscopía de e ec o únel espalda es a in e p e ación, con i mando
el compo amien o me álico del 1T-TaSSe y ayudando a la in e p e ación
expe imen al de la coexis encia de CDW y supe conduc i idad en es e sis-
ema.
Finalmen e, el úl imo capí ulo de la Tesis e lexiona sob e la madu ez y
las limi aciones de los ma cos eó ico-compu acionales desa ollados. En
el caso de los sis emas cuasi-1D, aunque se log a econcilia disc epancias
epo adas en la li e a u a, se iden i ica la necesidad de es ablece c i e ios
p edic i os sis emá icos pa a la búsqueda de nue os ma e iales. En cuan o
a las he e oes uc u as bidimensionales, el ma co p opues o demues a se
obus o, aunque se p e ee que su aplicación a sis emas de mayo compleji-
dad es uc u al eque i á ajus es adicionales. En ambos casos, los mé odos
desa ollados mues an po encial de escalabilidad y o ecen una base sóli-
da pa a ex ende su aplicación a nue as líneas de in es igación en ísica de
ma e iales.
En esumen, es a Tesis plan ea un en oque sis emá ico pa a el es udio de
las p opiedades elec ónicas de sis emas o mados po me ales de ansi-
ción con dimensionalidad educida. Es e en oque se lle a a cabo a a és
de la combinación de cálculos basados en DFT, écnicas de band un olding,
modelos de enlace ue e y alidación expe imen al colabo a i a, cons u-
yendo así una pla a o ma me odológica que no solo con ibuye a pa icipa
y a a de escla ece deba es abie os en el campo, sino que ambién ab e
nue as ías pa a la explo ación de ma e iales con p opiedades elec ónicas
exó icas y que albe gan enómenos elacionados con la ue es co elacio-
nes.
Pa I
INTRODUCTION
1
INTRODUCTION
The esea ch p esen ed in his Thesis lies a he in e sec ion o heo e i-
cal, compu a ional and expe imen al condensed ma e physics. Th ough-
ou he ollowing pages, ou goal is o de elop obus heo e ical ame-
wo ks ha se e as pla o ms o uni y, unde s and and con ex ualize pas
esea ch, while pa ing he way o u u e ad ancemen s in he ield. How-
e e , hese amewo ks canno exis in a acuum; hey a e i mly g ounded
in a b oade scien i ic and his o ical con ex . This in oduc ion aims o es-
ablish exac ly ha : he who,wha ,whe e, and when o his wo k1, se ing
he s age o he ques ions i add esses and he esea ch i aims o inspi e.
Fi s , we will p o ide a b ie quasi-c onological ou line o he imeline o
he de elopmen o condensed ma e physics as a ield, highligh ing he
disco e ies ha a e mos ele an o ou esea ch: he echniques employed
and he ongoing deba es in he ield whe e ou wo k can con ibu e. Follow-
ing his, we will p esen he s a e-o - he-a con ex in which ou esea ch
akes place, along wi h a b ie ou line connec ing ou wo k o he cu en
ad ancemen s in he ield. Finally, we will o e a b ie commen on ou
app oach.
1.1 his o ical and concep ual backg ound o his wo k
Condensed ma e physics is he b anch o physics ha s udies he mac o-
scopic and mic oscopic p ope ies o ma e in i s solid and liquid s a es,
whe e high pa icle densi ies make in e ac ions be ween cons i uen ions
and elec ons cen al o unde s anding hei beha io .
As a esul , condensed ma e physics ocuses on how la ge ensembles
o in e ac ing pa icles gi e ise o eme gen 2collec i e phenomena: beha -
io s ha canno be deduced om s udying he indi idual componen s in
isola ion. In wha ollows, we will ou line key disco e ies ha led o he
de elopmen o he cen al echniques and concep s ele an o his Thesis:
1These ques ions can be unde s ood in e ms o he " i e Ws" checklis . We ha e excluded
he why, as i pe ains o inal causes, which all ou side he scope o his Thesis. Addi ion-
ally, he ques ion o whe e will be add essed in e ms o a concep ual loca ion, a he han
a spa ial one.
2Eme gence occu s when a complex sys em (composed o mul iple en i ies, such as a oms
in he case o condensed ma e physics) exhibi s p ope ies o beha io s ha a e no
p esen in i s indi idual componen s, bu a ise h ough hei in e ac ions.
2
1.1 his o ical and concep ual backg ound o his wo k 3
p ima ily densi y unc ional heo y (DFT)3, bu also se e al many-body
phenomena ha , while no he p ima y ocus, o m an essen ial pa o he
concep ual backg ound o his wo k, such as cha ge densi y wa es (CDWs)
(Sec. 3.1.3), supe conduc i i y (App. C.9), and he Kondo and Mo e ec s
(Sec. 4.2.1).
To do so, we begin by e isi ing he ea ly classical ounda ions o con-
densed ma e physics and explo ing how he ad en o quan um mechan-
ics in he ea ly 20 h cen u y ans o med he ield. We illus a e his ans o -
ma ion h ough a selec ion o ep esen a i e disco e ies, which highligh
he powe o he quan um o malism. F om he e, we shi ou ocus o he
his o ical and concep ual mo i a ions unde lying he esea ch p esen ed
in Chap e s 3and 4, along wi h he echniques employed and he cu en
s a e-o - he-a in he ield.
1.1.1Fi s classical s eps.
To p o ide p ope empo al con ex o ou esea ch, we will ace back
o he i s known model o elec ical conduc ion4, he D ude model. The
choice o D ude’s model as he s a ing poin o ou sui gene is his o ical
e iew s ems om ou iew ha i ep esen s one o he ea lies a emp s
o unde s and he mac oscopic p ope ies o solids om a o mal heo e i-
cal pe spec i e, mo ing beyond pu ely phenomenological explana ions.
In 1900, Paul D ude p oposed a heo e ical model o elec onic conduc ion
based on classical physics [2]. In simple e ms, unde D ude’s model, me -
als a e desc ibed as a gas o ee elec ons mo ing h ough a pe iodic la ice
o ions, sca e ing o hem in a manne analogous o a pinball bouncing
o obs acles, he eby gi ing ise o esis i i y. Despi e i s limi a ions, his
model was ema kably success ul o i s ime, o e ing he i s mic oscopic
in e p e a ion o mac oscopic anspo laws in solids. I accoun ed o se -
3In sho , DFT is a quan um-mechanical amewo k o desc ibing many-elec on sys ems
based on hei g ound-s a e elec on densi y a he han hei ull many-body wa e unc-
ion. Founded on he Hohenbe g–Kohn heo ems and p ac ically implemen ed h ough
he Kohn–Sham scheme, DFT p o ides he basis o widely used compu a ional me hods
ha can p edic s uc u al, elec onic, and ene ge ic p ope ies o ma e ials wi h ema k-
able accu acy. We will mo i a e and p esen DFT in Chap e 2.
4His o ically, o he s udies can be ela ed o condensed ma e physics, such as he in es-
iga ions o condensed s a es ca ied ou by Humph y Da y o Michael Fa aday du ing
he nine een h cen u y. Howe e , we shall begin in 1900 wi h D ude’s model o conduc-
ion, as i p o ides a na u al s a ing poin o building up owa ds quan um ea men s
and highly co ela ed many-body phenomena. I is also he s a ing poin om "Solid
s a e physics" by Asch o & Me min [1], one o he go- o book o in oduc o y cou ses in
solid-s a e physics in he unde g adua e le el.
1.1 his o ical and concep ual backg ound o his wo k 4
e al empi ical obse a ions, including he Wiedemann–F anz law5[3].
Howe e , i was no wi hou sho comings: i signi ican ly o e es ima ed
he elec onic hea capaci y o me als and ailed o accoun o he em-
pe a u e dependence o esis i i y a low empe a u es, disc epancies ha
would la e mo i a e he inco po a ion o quan um mechanics in o solid-
s a e heo y.
1.1.2Ad en o quan um mechanics.
Wi h he ad en o quan um mechanics in he 1920s and 1930s, D ude’s
model and condensed ma e physics as a ield was p o oundly e ined by
physicis s such as W. Pauli, A. Somme eld, and F. Bloch; who employed
he eme ging quan um o malism o de elop mo e accu a e6 heo ies o
physical phenomena.
Fo ins ance, in 1927, Pauli ecognized ha elec ons in me als obey Fe mi–
Di ac s a is ics7and used his insigh o de elop a quan um heo y o pa a-
magne ism8[6]. Tha same yea , Somme eld inco po a ed Fe mi–Di ac
s a is ics in o he classical D ude model, gi ing ise o he ee-elec on
model, an impo an concep ual b eak h ough ha ma ked he ansi ion
om classical o quan um desc ip ions o elec ons in solids. By ea ing
elec ons as a degene a e Fe mi gas a he han a classical ensemble, his
model esol ed key disc epancies in he D ude heo y and p o ided a
signi ican ly imp o ed accoun o cha ge ca ie beha io in me als. No-
ably, i succeeded in explaining se e al expe imen al obse a ions, includ-
ing he ma kedly lowe elec onic hea capaci y o me als. Howe e , i s ill
neglec ed elec on–elec on in e ac ions, limi ing i s ange o applicabili y
and highligh ing he need o mo e sophis ica ed many-body ea men s,
which we will explo e la e .
5The Wiedemann–F anz law s a es ha he a io be ween he elec onic con ibu ion o
he he mal conduc i i y (κ) and he elec ical conduc i i y (σe) is p opo ional o he
empe a u e (T), o mally exp essed as κ/σe=KT, whe e Kis a cons an .
6By mo e accu a e, we mean ha hese heo ies o e ed g ea e p edic i e powe and showed
imp o ed ag eemen wi h expe imen al obse a ions in condensed ma e sys ems. Fo
ins ance, Somme eld’s quan um e inemen o D ude’s model success ully accoun ed o
he elec onic hea capaci y o me als, a key phenomenon ha he classical D ude model
could no explain, he eby ende ing i a mo e accu a e desc ip ion o me allic beha io .
7Fe mi–Di ac s a is ics, de i ed independen ly by E. Fe mi and P. Di ac a ound 1926 [4,5],
desc ibe sys ems o non-in e ac ing, iden ical e mions wi h hal -in ege spin ha obey
he Pauli exclusion p inciple.
8Pa amagne ism is a magne ic phenomenon in which pa amagne ic ma e ials a e weakly a -
ac ed o ex e nally applied magne ic ields and de elop an in e nal induced ield aligned
wi h he applied ield. In con as , diamagne ic ma e ials a e epelled by magne ic ields
and gene a e ields in he opposi e di ec ion.
1.1 his o ical and concep ual backg ound o his wo k 5
Meanwhile, in he la e 1920s and ea ly 1930s, ano he c ucial a enue o con-
densed ma e physics eme ged: c ys allog aphy. De eloped by A. B a ais
and Y. Fyodo o , c ys allog aphy p o ided essen ial ools o he s uc u al
analysis o c ys alline ma e ials9, leading o hei classi ica ion by symme-
y g oups and culmina ing in he publica ion o he In e na ional Tables o
C ys allog aphy in 1935. The impo ance o his de elopmen o condensed
ma e physics canno be o e s a ed: i o e ed a igo ous classi ica ion o
solids based on hei symme y, es ablished a b idge be ween a omic-scale
s uc u e and mac oscopic p ope ies, and laid he g oundwo k o band
heo y. A ound he same ime, in 1929, Bloch o mula ed his eponymous
heo em [7] (see Sec. 2.5), which became cen al o bands uc u e heo y. In
1930, L. B illouin in oduced he concep o he B illouin zone, p o iding
a compac amewo k o desc ibing wa e-like solu ions in pe iodic media.
Acco ding o Bloch’s heo em, all such solu ions can be ully cha ac e -
ized wi hin a single B illouin zone. C ys allog aphy, Bloch’s heo em, and
he B illouin zone emain o pa amoun impo ance in condensed ma e
physics in gene al, and in his Thesis in pa icula , as hey o m he concep-
ual and compu a ional ounda ion o bands uc u e calcula ions. Wi hou
a well-de ined c ys al s uc u e, one canno de ine he inpu s o DFT; and
wi hou Bloch’s heo em and B illouin zones, one canno compu e band-
s uc u es10.
DFT is he main compu a ional echnique employed in his Thesis. I was
i s o malized by W. Kohn and P. Hohenbe g in 1964 [8], bu i s concep-
ual o igins ace back o he Thomas–Fe mi model (1926), which desc ibed
elec ons in e ms o hei densi y and was u he e ined by Di ac in 1930
by including an analy ical exp ession o he exchange ene gy[9], key o
DFT. The de elopmen o DFT buil upon a numbe o key app oxima ions,
such as he Bo n–Oppenheime app oxima ion (1927) [10], Ha ee’s mean-
ield ea men (1928) [11], and he Ha ee–Fock gene aliza ion (1930) [12],
all o which a e discussed in Sec. 2.1. DFT has since ans o med condensed
ma e physics, enabling he disco e y o new ma e ials, he p edic ion o
opological phases, and he cons uc ion o obus compu a ional ame-
wo ks ha sys ema ize he s udy o ma e ial p ope ies 11.
9In he con ex o condensed ma e physics, a c ys al is a solid whose cons i uen a oms,
ions, o molecules a e a anged in a highly o de ed and epea ing pa e n ex ending in all
di ec ions.
10 A bands uc u e desc ibes he allowed ene gy le els o elec ons in a solid as a unc ion o
hei c ys al momen um. In a pe iodic la ice, Bloch’s heo em shows ha elec on wa e-
unc ions ake he o m o plane wa es modula ed by he la ice, leading o he o ma ion
o con inuous ene gy bands a he han disc e e a omic le els. The s uc u e and illing o
hese bands de e mine whe he a ma e ial beha es as a me al, semiconduc o , o insula o .
An example o a bands uc u e can be ound in Fig. 3.3
11 The decision o include a echnique such as DFT in he concep ual backg ound o his Thesis
is delibe a e: we unde s and heo y and echnique (whe he expe imen al, heo e ical, o
1.1 his o ical and concep ual backg ound o his wo k 6
Al hough DFT is he p ima y ool used in his Thesis, i is no he only
heo e ical echnique wo h con ex ualizing. In 1928, as pa o his doc-
o al disse a ion, F. Bloch in oduced he Linea Combina ion o A omic
O bi als (LCAO) me hod o solids12, o e ing an in ui i e app oach o
cons uc ing elec onic s a es om localized a omic o bi als. While pow-
e ul, his me hod was la e complemen ed by a mo e p ac ical and widely
used app oxima ion: he pa ame ized igh -binding model, de eloped by
J. C. Sla e and G. F. Kos e a ound 1954 [14]. This me hod is especially
e ec i e o s udying low-ene gy phenomena in complex ma e ials and un-
de pins he igh -binding models we use in Sec. 3.2.3, whe e we desc ibe
he quasi-1D beha io o (TaSe4)3I and (NbSe4)3I. Ano he indispensable
echnique in mode n compu a ional condensed ma e is Wannie iza ion
(see Sec. 2.7), which elies on he cons uc ion o localized Wannie unc-
ions, as in oduced by G. Wannie in 1937 [15].
While band heo y and i s compu a ional implemen a ions ha e success-
ully explained many aspec s o elec onic beha io , hey emain app oxi-
ma ions ha do no ully accoun o elec on–elec on in e ac ions. Fo in-
s ance, s anda d DFT includes elec on–elec on in e ac ions in an a e age
sense h ough he exchange-co ela ion unc ional, bu o en ails o cap u e
s ong co ela ion e ec s, such as hose a ising in sys ems wi h pa ially
illed do o bi als. These many-body in e ac ions gi e ise o a a ie y
o collec i e elec onic s a es ha go beyond e ec i e single-pa icle pic-
u es: phenomena such as Mo insula o s (see Sec. 4.2.1.1), i s p oposed
by N. F. Mo in 1937; supe conduc i i y, disco e ed in 1911 by H. K. Onnes
and F. Hols and la e o malized mic oscopically by Ba deen, Coope , and
Sch ie e in 1957 (see App. C.9); and he Kondo e ec , obse ed expe i-
men ally in he 1960s and explained by J. Kondo in 1964 (see Sec. 4.2.1.2).
Ano he collec i e phenomenon o pa icula ele ance o his Thesis is
he Cha ge Densi y Wa es (CDW), i s desc ibed by R. Peie ls in 1955 o
one-dimensional me als (see Sec. 3.1.3). CDWs we e among he ea lies mo-
i a ions o he s udy o low-dimensional sys ems, a subjec we now u n
o.
One key aspec o he sys ems s udied in his Thesis is ha mos o hem
exhibi e ec i e o eal low-dimensionali y. Dimensionali y has been a opic
o deba e in bo h physical sciences and philosophy since he ime o A is-
o le (384-322 BC). In mode n physics and ma hema ics, we can ace he
de elopmen o his concep o B. Riemann, who, in 1854, s udied highe -
compu a ional) o be linked by a coope a i e ela ionship in which one enables and shapes
he o he , see Sec. 1.3and App. A.1.
12 The me hod was i s in oduced o molecules by B. N. Finkles ein and G. E. Ho owi z,
and was hea ily in luenced by R. Mulliken’s wo k on molecula o bi als [13].
1.1 his o ical and concep ual backg ound o his wo k 7
dimensional mani olds, in luencing bo h geome y and heo e ical physics.
Following his, A. Eins ein’s Gene al Rela i i y (1916) combined space and
ime in o a ou -dimensional con inuum, while T. Kaluza (1921) and F.
Klein (1926) ex ended his amewo k o i e dimensions in an a emp o
uni y he undamen al o ces. Mo e ecen ly, s ing heo ies ha e sugges ed
he possibili y o 10 o 11 dimensions13 ( o example, M. G een in 1987
and J. Polchinski in 1998). In con as , condensed ma e physics is usually
mo e conce ned wi h he s udy o educed dimensionali y. The in e es in
dimensionali y can be aced back o he concep o quan um con inemen
de eloped in he 1920s and 1930s, which is closely linked wi h Bloch’s
heo em and band heo y. By he mid-20 h cen u y, heo e ical and expe -
imen al ounda ions we e es ablished o explo e genuine low-dimensional
ma e ials. This is he pe iod when, in he 1950s, R. Peie ls and H. F öhlich
p edic ed ha one-dimensional me als should spon aneously dis o in o
insula ing s a es, accompanied by pe iodic modula ions o he cha ge den-
si y, an e ec closely ela ed o CDWs. Expe imen al s udies on quasi-one-
dimensional (quasi-1D) conduc o s, such as ansi ion me al chalcogenides,
began in he 1970s and 1980s (ca ied ou by he g oups o A. Mee schau ,
P. G essie , and M. Izumi, among o he s [16–18]), cons uc ing he ounda-
ion o he esea ch p esen ed in Chap e 3. This esea ch aimed o iden i y
signa u es o Peie ls ansi ions and CDW s a es. Concu en ly, esea ch
in o laye ed wo-dimensional ma e ials14 pa icula ly g aphi e in e cala ion
compounds, gained ac ion, pa ing he way o u u e in es iga ions in o
mo e complex wo-dimensional sys ems.
A majo u ning poin occu ed in 2004, when he g oundb eaking isola ion
o g aphene ( i s heo ized by P. R. Wallace in 1947 [21]) by A. Geim and K.
No oselo [22] spa ked a global wa e o in e es in wo-dimensional ma e-
ials. This Nobel-winning disco e y led o ex ensi e s udies on elec onic
co ela ions, quan um anspo , and opological phenomena in a omically
hin laye s. Sho ly he ea e , a en ion shi ed apidly owa d ansi ion
me al dichalcogenides (TMDs), which eme ged as e sa ile, easily ex oli-
able, and highly unable ma e ials. Thei i s disco e y da es back o 1923
by L. Pauling [23], and he i s epo ed p oduc ion o MoS2monolaye
suspensions was in 1986. Seminal wo ks om he 2010s demons a ed ex-
cep ional elec onic, op ical and spin- alley physics in TMDs [24–29], e-
ealing phenomena such as indi ec - o-di ec bandgap ansi ions, supe -
conduc i i y, CDWs, and Mo insula ing s a es in he wo-dimensional
limi [30,31]. Mo e ecen ly, TMDs ha e e ol ed in bo h complexi y and
13 In s ing heo ies, ex a spa ial dimensions a e ypically compac i ied o cu led up a scales
smalle han a omic dimensions, ende ing hem unobse able a mac oscopic le els bu
s ill in luen ial in he undamen al laws o physics.
14 Ini ially conside ed he modynamically uns able and non-exis en by L. Landau [19] and
R. Peie ls [20].
1.2 s a e-o - he a ,goals and ou line o his wo k 8
scien i ic in e es , especially wi h he enginee ing o TMD he e os uc u es
[32]. In pa icula , he e obilaye s and wis ed mul ilaye s ha e c ea ed new
pla o ms o s udying s ongly-co ela ed phenomena. This is he esea ch
domain explo ed in Chap e 4.
1.2 s a e-o - he a ,goals and ou line o his wo k
As men ioned in he p eamble o his in oduc ion, he p ima y goal o
his Thesis is o de elop obus heo e ical amewo ks ha se e as pla -
o ms o uni y, unde s and, and con ex ualize bo h pas and u u e e-
sea ch. These amewo ks a e embedded wi hin he scien i ic and socio-
logical landscapes om which hey eme ge, making his Thesis a p oduc
o i s ime, shaped by con empo a y deba es in condensed ma e physics
and in o med by he disco e ies and con ibu ions o ou closes pee s.
Wha binds he di e en chap e s o his wo k oge he , beyond a sha ed
me hodology, is he consis en ocus on quan um ma e ials composed o
ansi ion me als wi h a ying oxida ion s a es. These a ia ions c ucially
a ec he ma e ials’ elec onic s uc u e and collec i e beha io , o e ing a
ich es ing g ound o he heo e ical ools we de elop. In his Sec ion, we
aim o con ex ualize he open ques ions, answe s, and esul s p esen ed in
his Thesis bo h his o ically and concep ually, highligh ing how ou con i-
bu ions in e sec wi h and ad ance he s a e-o - he-a in condensed ma e
physics.
Chap e 2is de o ed o a concep ual and me hodological e iew o he
many-body Hamil onian in condensed ma e physics, along wi h he ap-
p oxima ions and compu a ional echniques commonly used o add ess
i s inhe en complexi y. We begin by in oducing he gene al o m o he
Hamil onian, ollowed by an explo a ion o he main app oxima ions ha
ende he p oblem ac able. These include he Bo n–Oppenheime ap-
p oxima ion in Sec ion 2.2.1, as well as he Ha ee and Ha ee–Fock ap-
p oxima ions in Sec ions 2.2.2and 2.2.3, espec i ely. Subsequen ly, we e-
iew and analyze he ad en o DFT in Sec ion 2.3, examining i s heo e ical
ounda ions, compu a ional cos , and key app oxima ions: such as he Lo-
cal Densi y App oxima ion (LDA) and he Gene alized G adien App ox-
ima ion (GGA), discussed in Sec ions 2.3.3and 2.3.4. To conclude he dis-
cussion on DFT, we in oduce he pseudopo en ial echnique in Sec ion 2.4,
which signi ican ly educes compu a ional cos by eplacing he explici
ea men o co e elec ons wi h e ec i e ionic po en ials. This simpli ica-
ion is essen ial o e icien ly s udying he sys ems add essed in his Thesis.
Finally, we in oduce Bloch’s heo em in Sec ion 2.5, along wi h Wannie -
based echniques in Sec ion 2.7, which p o ide a powe ul b idge be ween
i s -p inciples calcula ions and e ec i e igh -binding models.
Pa II
THEORY & METHODS
2
UNRAVELING THE MANY-BODY ELECTRONIC
HAMILTONIAN
Du ing he in oduc ion o his Thesis, we con ex ualized ou esea ch
wi hin he b oade landscape o scien i ic knowledge o which i belongs.
Beyond i s de ining coo dina es wi hin his landscape, he cohesion o his
manusc ip is no solely de i ed om he simila i ies among he s udied
sys ems bu a he om he app oach aken o in es iga e hem. As s a ed
in he in oduc ion, he wo k p esen ed in he ollowing pages aspi es o
build a obus and sys ema ic unde s anding o hese ma e ials. To ha
end, we aim o de elop heo e ical amewo ks ha no only uni y pas e-
sea ch bu also lay a solid ounda ion o u u e heo e ical and expe imen-
al ad ancemen s. To achie e his, we p ima ily employ ab ini io me hods,
wi h a pa icula ocus on DFT. DFT se es as an ideal ool o his sys-
ema ic app oach due o i s heo e ical igo and compu a ional eliabili y.
The majo i y o he heo e ical esul s p esen ed in his Thesis a e de i ed
wi hin he DFT amewo k and, he e o e, will be conside ed co ec 1as
long as he app oxima ions inhe en o DFT emain applicable. This Chap-
e is dedica ed p ecisely o unde s anding he ounda ions, limi a ions,
and compu a ional echniques associa ed wi h DFT. We begin by o mula -
ing he mos gene al many-body Hamil onian o a sys em o in e ac ing
elec ons and ions wi hin quan um mechanics, g adually de eloping he
co e p inciples and app oxima ions ha de ine DFT. In doing so, we will
highligh i s compu a ional scaling, physical app oxima ions, and he ech-
niques ha e ine and ex end i s applicabili y such as Wannie unc ions.
This s uc u ed app oach will se e as he me hodological ounda ion o
he esul s p esen ed in he subsequen Chap e s.
As we an icipa ed du ing he in oduc ion, he esea ch p esen ed in his
Thesis is mainly loca ed wi hin he ield o condensed ma e physics. In
said ield, he low-ene gy beha io o ma e ials is mainly go e ned by he
in e ac ions be ween elec ons and ions in a many-body quan um sys em.
While, heo e ically, such sys ems can be ully desc ibed by a gene al many-
body Hamil onian, he deep complexi y a ising om he as numbe o
in e ac ing pa icles makes an exac solu ion compu a ionally in ac able.
Consequen ly, a hie a chy o app oxima ions is equi ed o make p og ess
in unde s anding hese sys ems.
This Chap e begins by o mula ing he many-body Hamil onian o elec-
1Aligning wi h commen p esen ed in Sec. 1.3and App. A.1
16
2.1 a hamil onian o gobe n hem all 17
onic sys ems and sys ema ically in oducing he app oxima ions neces-
sa y o make he p oblem compu a ionally ac able2. S a ing wi h unda-
men al quan um mechanical p inciples, such as he Bo n-Oppenheime
app oxima ion, we p og essi ely e ine he ea men o elec onic in e ac-
ions h ough he Ha ee and Ha ee-Fock me hods, ul ima ely a i ing a
DFT and he pseudopo en ial app oxima ion. Each s ep in his me hodolog-
ical p og ession educes he compu a ional complexi y while main aining
accu acy by inco po a ing physically meaning ul app oxima ions ailo ed
o he cha ac e is ics o he s udied sys ems. The culmina ion o hese de-
elopmen s is DFT, which p o ides an e icien and accu a e amewo k
o desc ibing he elec onic s uc u e o condensed ma e sys ems. I s bal-
ance be ween compu a ional easibili y and p edic i e powe has made i
he co ne s one o mode n compu a ional condensed ma e physics. As
such, DFT se es as he p incipal heo e ical ool employed h oughou
his Thesis, unde pinning he sys ema ic app oach aken o in es iga e he
ma e ials discussed in he ollowing Chap e s.
2.1 a hamil onian o gobe n hem all
In he con ex o he Quan um Mechanics, he Hamil onian His an ope a o
ha ep esen s he o al ene gy o a quan um sys em. The ime e olu ion
o such sys em is go e ned by he Sch ödinge equa ion, which can be
exp essed as
i
h∂
∂ |ΨS( )⟩=H |ΨS( )⟩, (2.1)
whe e |ΨS( )⟩is he quan um s a e o he sys em in Di ac no a ion.
In condensed ma e sys ems, which a e he p ima y ocus o his Thesis,
|ΨS⟩desc ibes he beha iou o all pa icles in he sys em: basically elec-
ons and ions. S a ing om he Hamil onian o mula ion, we can exp ess
he Hamil onian ope a o as
H=ˆ
T+ˆ
V, (2.2)
whe e ˆ
Tand ˆ
Va e he kine ic and po en ial ene gy ope a o s o he sys em,
espec i ely. In his manusc ip , all he sys ems o s udy a e cons i u ed
2In ligh o he ideas discussed in Sec. 1.3and App. A.1, when we say a p oblem is "com-
pu a ionally ac able," we don’ mean he e’s some ul ima e u h we’ e ailing o each
because ou compu e s a en’ as enough. Ra he , we ecognize ha he ools and ech-
niques a ailable o us de ine he bounda ies o wha kinds o ope a ions we can meaning-
ully ca y ou in condensed ma e physics oday. App oxima ions, hen, a e no laws o
sho cu s in ou heo ies, bu pa o he legi ima e se o ope a ions we ely on o gene a e
meaning ul and ep oducible esul s. They make i possible o cons uc he esul s p e-
sen ed in his Thesis, which is why we conside hem no only necessa y, bu undamen al
o he wo k included in his Chap e .
2.1 a hamil onian o gobe n hem all 18
by ions and elec ons in e ac ing wi h each o he 3. In o de o cap u e all
ele an in e ac ions in hese sys ems, we can ew i e 2.2as
H=ˆ
TI+ˆ
Te+ˆ
VI−I+ˆ
Ve−e+ˆ
Ve−I, (2.3)
whe e he kine ic ope a o s om ions and elec ons a e deno ed by ˆ
TIand
ˆ
Te, espec i ely. In 2.3, he in e ac ion e ms wi hin ions and wi hin elec-
ons a e cap u ed by ˆ
VI−Iand ˆ
Ve−ewhile ˆ
Ve−I ep esen s he in e ac ion
be ween elec ons and ions. We can u he expand 2.3as
H=−
h2
2meX
i∇2
i−X
i,I
ZIe2
| i−RI|+1
2X
i=j
e2
| i− j|
X
I
h2
2MI∇2
I+X
I=J
1
2
ZIZJe2
|RI−RJ|, (2.4)
whe e
his he Plank’s cons an and he e ms ela ed o elec ons a e de-
no ed by lowe case subsc ip s and ions, wi h cha ge ZIand mass MI, a e
deno ed by uppe case subsc ip s. In 2.4,i uns o e he numbe o elec-
ons Neand I uns o e he numbe o ions NI.
The exp ession in 2.4 ep esen s he simples o m o he Hamil onian o a
condensed ma e sys em, accoun ing solely o in e ac ions be ween he
pa icles wi hin he sys em i sel , wi hou ex e nal magne ic o elec ic
ields. I migh seem in ui i e ha , gi en all he necessa y in o ma ion,
one could p edic he sys em’s p ope ies and ime e olu ion by sol ing
he Sch ödinge equa ion o he en i e sys em.
HΨS({ i,σi;RI}; ) = ESΨS({ i,σi;RI}; ), (2.5)
whe e ΨSand ES ep esen he wa e unc ion4and o al ene gy o he en-
i e condensed ma e sys em. Howe e , he e a e p ac ical easons why
his app oach is no easible, highligh ing he need o app oxima ions o
sol e such Hamil onians, especially hose in ol ing mo e han wo pa i-
cles. In he nex ew pa ag aphs we add ess he wo p ima y sou ces o
hese limi a ions5:
1. I is a many-body p oblem o , a leas , EXP compu a ional complex-
i y [61].
3i.e. elec ons wi h elec ons, elec ons wi h ions and ions wi h ions.
4Jus a b ie no e on no a ion: p e iously, we we e using he Di ac no a ion o deno e he
quan um s a e o he whole sys em |ΨS⟩and making explici i s ime dependence. In Di ac
no a ion, he s a e is jus a ec o in he Hilbe space spanned by he eigens a es om H.
Now we will use he wa e unc ion o con enience, which is jus he ep esen a ion o he
s a e in he space-basis. Usually we can ela e bo h by ψ(x) = ⟨x|ψ⟩.
5Ac ually, bo h sou ces o limi a ions a e e y simila and could be ea ed as one. None he-
less, he choice o make dis inc ion in en ionally lea es oom o how quan um compu a ion
could ad ess co ela ion e ec s in a much mo e e icien ashion h ough he secod poin .
2.2 app oxima ions 19
The Hamil onian in 2.4desc ibes a many-body sys em wi h in e ac-
ions among elec ons, among ions, and be ween elec ons and ions.
The wa e unc ion o his sys em depends on he posi ions o all elec-
ons and ions, so o N=Ne+NIpa icles, he wa e unc ion is de-
ined o e a 3N-dimensional con inuous space (wi hou conside ing
addi ional in e nal deg ees o eedom such as spin),
Ψ:R3N →C. (2.6)
To ea he p oblem nume ically, one mus disc e ize his space using
Mg id poin s pe spa ial coo dina e, leading o a o al o M3N possi-
ble con igu a ions. As a esul , bo h s o ing he wa e unc ion and pe -
o ming exac diagonaliza ion o he Hamil onian equi e compu a-
ional esou ces ha scale exponen ially wi h N. This places he many-
body p oblem, in i s ull gene ali y, wi hin he class o p oblems o
a leas EXP compu a ional complexi y 6, ende ing i in ac able o
la ge N.
2.Co ela ion e ec s mus be aken in o accoun .
The hi d e m in he Hamil onian 1
2Pi=je2
| i− j|in oduces s ong
co ela ions among elec ons. Each elec on expe iences he in luence
o all o he elec ons, leading o a complex many-body p oblem. Elec-
ons obey he Pauli exclusion p inciple and he e ec s o exchange
in e ac ion, which a ises om hei indis inguishabili y and he e-
que imen o hei o al wa e unc ion o be an i-symme ic unde pa -
icle exchange. The co ec ea men o his in e ac ions canno be
a ained simply by applying pai wise o ces as in a classical con ex .
Ins ead, he solu ion would in ol e non-local in e ac ions and en an-
glemen , whe e he s a e o one pa icle depends on he s a e o all
o he s. These quan um e ec s canno be e icien ly simula ed by a
classical compu e due o he exponen ial scaling p oblem coming
om quan um phenomena.
2.2 app oxima ions
While he challenges p esen ed in he p e ious Sec ion migh seem discou -
aging, hey mo i a e ou need o de elope app oxima ions ha make he
6A p oblem is EXP-comple e i i can be sol ed in de e minis ic exponen ial ime and is
EXP-ha d, meaning ha any o he p oblem in he class EXP can be educed o i us-
ing polynomial- ime ans o ma ions. In he case o many-body sys ems, he exponen ial
g ow h o he Hilbe space wi h pa icle numbe implies ha bo h s o age and spec al
que ies (e.g., whe he he g ound s a e ene gy lies below a gi en h eshold) equi e expo-
nen ial ime, and a e among he ha des p oblems in EXP.
2.2 app oxima ions 20
p oblem ac able. We will begin by educing he e ec i e pa icle coun ,
N, inside he big O no a ion O(M3N).
2.2.1Bo n-Oppenheime app oxima ion
The i s s ep in simpli ying ou p oblem is o decouple he ionic and elec-
onic deg ees o eedom. Kine ic ene gy e ms in he Hamil onian a e
linea ly p opo ional o pa icle mass and quad a ically p opo ional o e-
loci y. Acco ding o equipa i ion heo em7bo h o hese e ms will be also
p opo ional o 3/2kBT, implying ha he a io o hei eloci ies will scale
as
I
e
= me
mI
. (2.7)
F om 2.7and by subs i u ing he mass o he ligh es nucleus we can ind,
i.e. hyd ogen wi h MI≈1836 me, we can conclude ha ions mo e a leas
a ound 40 imes slowe han elec ons. Consequen ly, on he ime-scale o
nuclea (ionic) mo ion, elec ons a e expec ed o apidly elax o an in-
s an aneous g ound-s a e con igu a ion. This will allow us o app oxima e
he ions as s a iona y pa icles when ob aining he elec onic g ound-s a e
by sol ing he ime-independen Sch ödinge equa ion, an app oxima ion
which is known as he Bo n-Oppenheime (BO) app oxima ion [10].
In o de o p ope ly in oduce he BO app oxima ion, le s s a by s a ing
he Adiaba ic heo em,
Theo em 2.2.1A physical sys em emains in i s ins an aneous eigens a e i a
gi en pe u ba ion is ac ing on i slowly enough and i he e is a gap be ween
he eigen alue and he es o he Hamil onian’s spec um [62].
Then, he BO app oxima ion assumes ha elec onic eigens a es a e no
a ec ed by pe u ba ions o mo emen s om he ions. To alida e his as-
sump ion, we ollow [63] and assume ha he wa e unc ion o he en i e
sys em, including elec ons and ions, can be exp essed as
˜
Ψ({ i},{RI}) = Ψ({ i};{RI})Φ({RI}), (2.8)
In exp ession 2.8, he wa e unc ion o he en i e sys em is di ided in o wo
componen s: Ψ({ i};{RI}), which ep esen s he elec onic pa in luenced by
7The equipa i ion heo em s a es ha , a he mal equilib ium, each deg ee o eedom
con ibu es 1
2kBT o he sys em’s a e age ene gy (in he main ex we a e using he a e age,
Ekin =m
2( 2
x+ 2
y+ 2
z)→ ⟨Ekin⟩=3/2kBT). E en hough his heo em is pa icula ly
sui ed o classical sys ems, since he quan um kine ic ene y ope a o scales in e sely wi h
mass −
h2
2m ∇2∝1
m, we can use i o s a e ha o pa icles in mo ion in a quan um sys em,
hei eloci y ∝q1
m.
2.2 app oxima ions 21
bo h he ionic and elec onic posi ions, and Φ({RI}), which ep esen s he
ionic pa and depends only on he ionic posi ions. This sepa a ion assumes
ha he wo componen s a e independen 8. Fu he mo e, we equi e ha
Ψ({ i};{RI})sa is ies ime-independen Sch ödinge equa ion,
1
2
X
iX
j=i
1
| i− j|−X
i∇2
i−X
i,I
2ZI
| i−RI|+
Ψ({ i};{RI})
=Ee(RI)Ψ({ i};{RI}), (2.9)
whe e we aknowledge he dependence o he elec onic ene gy eigen alue
(o adiaba ic ene gy con ibu ion) Ee(RI), on he ixed ionic posi ions9. Ap-
plying he ull Hamil onian 2.4 o he en i e wa e unc ion and se ing
h=me=e2=4πϵ0=1, (2.10)
we a i e a
H˜
Ψ({ i};{RI}) =
−X
j
1
2MJ∇2
I+Ee({RI})
+1
2X
JX
K=J
ZJZK
|RJ−RK|
˜
Ψ({ i},{RI})
=Ψ({ i};{RI})
−X
J
1
2MJ∇2
J+Ee({RI}
+1
2X
JX
K=J
ZJZK
|RJ−RK|
Φ({RI})
−X
J
1
2MJ
[2∇JΦ({RI})·∇JΨ({ i};{RI}) + Φ({RI})∇2
JΨ({ i};{RI})i,
(2.11)
whe e Ee({RI})is called he adiaba ic ene ge ic con ibu ion. The emaining
non-adiaba ic e ms con ibu e neglibly o he ene gy. To show his, we ap-
ply ime-independen pe u ba ion heo y. The i s o de co ec ion om
non-adiaba ic e ms has he o m 10
8He e we a e conside ing wo independen subsys ems whose wa e unc ions li e in a
Hilbe space ha is he enso p oduc o he indi idual Hilbe spaces o each sub-
sys em, since he sys ems a e independen , he enso p oduc s a e can be (and is) ep e-
sen ed by he p oduc o s a es. This can also be unde s ood as he join p obabili y o wo
independen e en s.
9Since he elec ons a e expec ed o apidly adap o a g ound-s a e depending on a se o
ixed a omic posi ions, he elec o nic ene gy eigen alue Ee(o wha we will call adiaba ic
ene ge ic con ibu ion la e ) only depends on said ionic posi ions.
10 He e ×is only used o indica e a line b eak ha sepa a es a scala p oduc , no he ec o-
ial p oduc .
2.2 app oxima ions 22
−ZY
j
d jY
β
dRJΨ∗({ i};{RI})Φ∗({RJ})×
X
K
1
MK
[∇KΦ({RI})·∇KΨ({ i};{RI})]
= − X
KZY
J
dRJΦ∗({RI})∇KΦ({RJ})×(2.12)
ZY
j
d jΨ∗({ i};{RI})∇KΨ({ i};{RI})
, (2.13)
whe e he e m in b acke s anishes,
ZY
j
d jΨ∗({ i};{RI})∇KΨ({ i};{RI})
=1
2∇KZY
j
d j|Ψ({ i};{RI})|2=1
2∇K(1) = 0, (2.14)
since he no malisa ion o he elec o nic wa e- unc ion does no change
when he ions mo e, making he i s -o de con ibu ion o anish. The
second-o de shi gi es ise o ansi ions in he elec ons when he ions
mo e, his phenomenom i s known as elec on-phonon in e ac ion, and has
an impac on he o al ene gy11. The second non-adiaba ic e m becomes
signi ican when he elec ons ia e igh ly bound o ions I. In ha case, we
can exp ess he wa e unc ion as depending on he dis ance α(i,I)be ween
elec ons and ions,
Ψ({ i};{RI}) = Ψ({α(i,I)}), (2.15)
whe e α(i,I)= i−RI. The i s -o de co ec ion o his e m is
−ZY
j
d jY
J
d J{α(i,I)}Φ∗({RI})X
K
1
2MKhΦ({ I})∇2
K{α(i,I)}i
=X
K
1
2"ZY
K
d J|Φ({RI}|2#
ZY
j,J
dα(j,J)Ψ∗{α(i,I)}∇2
KΨ{d(i,I)}
=X
(k,K)
1
MKZY
(j,J)
dα(j,J)Ψ∗{α(i,I)}1
2∇2
(k,K)Ψ{α(i,I)}, (2.16)
which is o he o de o he elec o nic kine ic ene gy mul iplied by he a io
me/mI, allowing us o ea i as negligible. Consequen ly, we can hen
11 This elec on-phonon pa is c ucial o all he phenomena su ounding he sys ems s ud-
ied in his Thesis, mo e speci ically hose in Chap e 4.
2.2 app oxima ions 23
neglec all non-adiaba ic e ms and decouple he Sch ödinge equa ion o
ions as
−X
J
1
2MJ∇2
J+Ee({RI}) + 1
2X
JX
K=J
ZJZK
| J− K|
Φ({ I})=EΦ({ I}), (2.17)
which allows us o ea he non-adiaba ic elec onic e ms independen ly.
This app oxima ion will d as ically educe Nin O(N3M) h ough he de-
coupling o ionic and elec o nic deg ees o eedom, p o iding a c ucial
simpli ica ion in he endea o o sol e he many-body Hamil onian.
2.2.2Ha ee app oxima ion
Thanks o he BO app oxima ion, we ha e succes ully simpli ied he ionic
+ elec onic p oblem in o a elec onic one by decoupling he mo ion o
elec ons and ions. Howe e , e en hough his has educed he e ec i e
numbe o pa icles in he p oblem12, we a e s ill dealing wi h a p oblem
o , a leas , EXP-comple e compu a ional complexi y. The nex logical s ep
u he in ou app oach o sol ing he Hamil onian is he me hod p oposed
by Douglas Ha ee [11] in 1928, which ea s he elec ons as independen
pa icles ha mo e independen ly wi hin an a e aged po en ial c ea ed by
all o he elec ons. This app oach e ec i ely educes he complexi y o he
p oblem in o a mo e manageable polynomial scaling o O(N).
The i s s ep in his me hod is wha is known as he independen pa icle
app oxima ion: In a many-body Hamil onian o Npa icles o spin σgi en
by
H( 1, 2,..., N; ) =
N
X
i=1
ˆ
P2
i
2mi
+Vσ( 1, 2,..., N; ), (2.18)
whe e ˆ
Pand mia e he momen um ope a o and mass o pa icle i. I
pa icles in 2.18 do no in e ac wi h each o he , we can exp ess he po en ial
e m as a sum o independen e ms
Vσ( 1, 2,..., N; ) =
N
X
i=1
Vσ(i)( i; ), (2.19)
and hence, ew i e he Hamil onian as
H( 1, 2,..., N; ) =
N
X
i=1
H( i; ). (2.20)
12 Nin O(N3M).
2.2 app oxima ions 24
Since he pa icles a e non-in e ac ing, we can ep esen he sys em’s wa e-
unc ion, Ψ, as a p oduc o each pa icle’s wa e unc ion ϕσ
i:
Ψ( 1, 2,..., N; ) = ϕσ
1( 1, )ϕσ
2( 2, )...ϕσ
N( N, ). (2.21)
Each o hese single-pa icle wa e unc ions ϕie ol es o e ime as
ϕi( i; ) = ϕϵi( i)e−iϵi
h, (2.22)
allowing he sys em’s wa e unc ion o be exp essed as
Ψ( 1, 2,..., n; ) = ΨE( 1, 2,..., n)e−iE
h, (2.23)
whe e E=PN
i=1ϵi.
I we combine bo h independen elec on app oxima ion and he BO ap-
p oxima ion, we a i e o he Ha ee app oxima ion [11]. By exp essing
2.21 as
ψ( ) =
N
Y
i
ϕσ
i( i), (2.24)
we can w i e an e ec i e Hamil onian o each single-pa icles wa e unc-
ions ϕσ
i( i)wi h spin σ,
He ϕσ
i( i) = −1
2∇2+Vσ(i)
e ( i)ϕσ
i( i) = ϵσ
iϕσ
i( i). (2.25)
In exp ession 2.25, we can see ha he po en ial has been ac o ized in an e -
ec i e po en ial, Vσ(i)
e ( i), which ac s on each elec on o spin σa posi ion
i. Due o he undamen al na u e o Ha ee app oxima ion, his po en ial
mus inco po a e he e ec s o all o he pa icles. In Ha ee app oxima ion
we assume ha each elec on in e ac s wi h e e y o he elec o n p opo ion-
ally o he p oduc o he densi ies o each pai as ollows,
⟨H(e)⟩ψ=X
i,σ⟨He ⟩ϕσ
i(2.26)
⟨H(e)⟩ψ=X
i,σ⟨ϕσ
i|−1
2∇2
i+Ve−I( i)|ϕσ
i⟩
+1
2X
i,σX
j,σ′=i,σ⟨ϕσ
i,ϕσ′
j|1
| i− j||ϕσ
i,ϕσ′
j⟩. (2.27)
Now, le s sol e his e ec i e Hamil onian by using he a ia ional heo em
(see App. B.1). To do his, we de ine a unc ional F(see App. B.2) wi h he
co esponding Lag ange mul iplie s ϵi
F[{ϕσ
i},{ϵi}]=⟨H⟩ψ−X
i
ϵi X
σ⟨ϕσ
i|ϕσ
i⟩−1!. (2.28)
2.3 densi y unc ional heo y 31
HK
⇒
⇒
⇒
⇒
Ψ0({ })Ψi({ })
n0( )Vex ( )
Figu e 2.1: Schema ic ep esen a ion o Hohenbe g-Kohn ho em. S a ing om
he uppe le side, he po en ial Vex ( )de e mines (⇒) all he s a es
Ψi({ })o he sys em, including he g ound-s a e (⇒)Ψ0({ })and hence
(⇒) he g ound-s a e densi y n0( ). The Hohenbe g-Kohn heo em
s a es ha Vex ( )is uniquely de e mined by n0( ), closing he ci cle.
I we hen add Eqs. 2.47 and 2.48, we a i e o
E(1)+E(2)< E(1)+E(2), (2.49)
which is a con adic ion ha ende s ou hypo hesis inco ec . Wi h his
p oo by con adic ion, we can conclude ha he e is no wo di e en po-
en ials which gi e ise o he same non-degene a e g ound-s a e cha ge
densi y n0. Hence, he densi y uniquely de e mines he ex e nal po en ial
wi hin a cons an .
Theo em 2.3.2A uni e sal unc ional o he ene gy E[n]in e ms o he
densi y n( )can be de ined, alid o any ex e nal po en ial Vex ( ). Fo any
pa icula Vex ( ), he exac g ound-s a e ene gy o he sys em is he global
minimum alue o his unc ional, and he densi y n( ) ha minimizes he
unc ional is he exac g ound-s a e densi y n0( ).
Co olla y 2.3.2The unc ional E[n]alone is su icien o de e mine he exac
g ound-s a e ene gy and densi y.
P oo . The p oo o he second heo em ollows a simila s uc u e o ha
one o he i s , wi h some addi ional conside a ions. This p oo , as he o ig-
inal om Hohenbe g-Kohn, is es ic ed o V- ep esen able densi ies, i.e. den-
si ies ha co espond18 o he g ound-s a e o an elec on Hamil onian wi h
some ex e nal po en ial Vex . Since all p ope ies a e uniquely de e mined
by n( ), hey can be exp essed as unc ionals o said densi y, including he
o al ene gy unc ional
EHK[n] = T[n] + Ein [n] + Zd Vex ( )n( ) + EII
=FHK[n] + Zd Vex ( )n( ) + EII, (2.50)
18 Can be de i ed as he g ound-s a e densi y o a physical many-body sys em.
2.3 densi y unc ional heo y 32
KS
⇔
{ϕi( )}
VKS( )
n0( )HK0
⇒
⇒
⇒
⇒
Φ( )
HK
⇒
⇒
⇒
⇒
Ψ0({ })
Ψi({ })
n0( )Vex ( )
Figu e 2.2: Schema ic ep esen a ion o Kohn-Sham app oach. HK0deno es
Hohenbe g-Kohn heo ems applied o he non-in e ac ing sys em. KS
bidi ec ional a ow signals he ela ion be ween he many-body and
single-pa icle sys ems, showing ha he solu ion o he independen -
pa icle Kohn-Sham p oblem de e mines he p ope ies om he ull
many-body sys em.
whe e T[n]and Ein [n]a e he unc ionals o he kine ic and po en ial en-
e gies espec i ely and EII is he ion-ion in e ac ion ene gy. In he second
line o Eq. 2.50 we w i e FHK[n], a uni e sal unc ional encompassing e ms
ha a e independen o he ex e nal po en ial.
To p o e he heo em, le s conside a sys em wi h g ound-s a e densi y
n(1)( )co esponding o he ex e nal po en ial V(1)
ex ( ). Using he g ound-
s a e wa e unc ion Ψ(1), he Hohenbe g-Kohn unc ional can be exp essed
as19
E(1)=EHK hn(1)i=⟨Ψ(1)| H(1)|Ψ(1)⟩. (2.51)
Now, conside a di e en densi y n(2)( )associa ed wi h a di e en wa e-
unc ion Ψ(2). Following he same logic as in he p oo o he i s heo em,
we can e o mula e Eq. 2.45 as
E(1)=⟨Ψ(1)| H(1)|Ψ(1)⟩<⟨Ψ(2)| H(2)|Ψ(2)⟩=E(2). (2.52)
This inequali y shows ha any densi y n(2)( )di e en om he g ound-
s a e densi y n(1)( )yields a highe o al ene gy. The e o e, he global min-
imum o FHK[n]co esponds o he ue g ound-s a e densi y. Since he
Hohnenbe g-Kohn unc ional FHK[n]is uni e sal and independen o Vex ( ),
he exac g ound-s a e denis y and ene gy cna be ound by minimizing he
o al ene gy wi h espec o a ia ions in n( ). This esul suppos s Co ol-
la y 2.3.2, es ablishing ha E[n]alone is su icien o de e mine he exac
g ound-s a e ene gy and densi y.
19 He e, we omi he explici dependence o non o simpli y he no a ion.
2.3 densi y unc ional heo y 33
2.3.2The Kohn-Sham app oach
While he Hohenbe g-Kohn heo ems s a e ha he g ound-s a e elec on
densi y n0( )de e mines all p ope ies o a sys em, hey do no p o ide a
s aigh o wa d me hod o ob aining n0( ). The Kohn-Sham app oach, de-
pic ed in Fig.2.2, ackles his p oblem by in oducing an auxilia y sys em o
non-in e ac ing elec ons, de ined o ha e he same g ound-s a e densi y as
he in e ac ing sys em. This ecas s he p oblem in o a se o sel -consis en
equa ions ha a e nume ically sol able, while e aining he complexi y o
he eal sys em h ough an exchange-co ela ion unc ional. This app oach
e ec i ely educes he many-body p oblem o an independen -pa icle one,
making i compu a ionally easible.
The cons uc ion o he Kohn-Sham auxilia y sys em is based in wo key
assump ions,
1. The exac in e ac ing g ound-s a e densi y, ne
0( ), is ep esen able by
he g ound-s a e densi y o an auxilia y non-in e ac ing sys em na
0( ).
2. The auxilia y Hamil onian Haux has he usual kine ic ope a o and
e ec i e local po en ial Vσ
e ( ), ac ing on an elec on o spin σa .
Based on hese assump ions, we can o mula e he auxilia y Hamil onian
as20
Hσ
aux = −1
2∇2+Vσ( ). (2.53)
Since he auxilia y Hamil onian, Hσ
aux om Eq. 2.53 is non-in e ac ing, he
g ound-s a e o he sys em is con igu ed by placing one elec on in each o
he Nσo bi als ψσ
i( )wi h he lowes eigen alues ϵσ
i. Hence, he elec on
densi y o he auxilia y sys em is de ined as
n( ) = X
σ
n( ,σ) = X
σ
Nσ
X
i=1
|ψσ
i( )|2. (2.54)
The he kine ic ene gy TSo he auxilia y sys em is gi en by
TS= −1
2X
σ
Nσ
X
i=1⟨ψσ
i|∇2|ψσ
i⟩=1
2X
σ
Nσ
X
i=1Zd |∇ψσ
i( )|2. (2.55)
And he classical Coulomb in e ac ion ene gy, esul ing om he elec on
densi y in e ac ing wi h i sel , is
EC[n] = 1
2Zd d ′n( )n( ′)
| − ′|. (2.56)
20 Using Ha ee uni s,
h=me=e=4π/ϵ0=1.
2.3 densi y unc ional heo y 34
Wi hin he Kohn-Sham app oach, he ull in e ac ing many-body Hohenbe g-
Kohn ene gy unc ional om Eq. 2.50 can be ew i en as
EKS =TS[n] + Zd Vex ( )n( ) + EC[n] + EII +Exc[n], (2.57)
whe e Vex ( )is he ex e nal po en ial owing o ions and ex e nal ields,
EII is he in e ac ion be ween ions, ECis he Coulomb ene gy de ined in
Eq. 2.56 and Exc is he exchange-co ela ion ene gy ha encapsula es all
many-body e ec s o exchange and co ela ion. The exchange-co ela ion
ene gy can be exp essed in e ms o he Hohenbe g-Kohn unc ional FHK[n],
Exc[n] = FHK[n] − (TS[n] + EC[n]) = ⟨T⟩−TS[n] + ⟨Ein ⟩−EC, (2.58)
whe e n( )is he densi y and depends upon posi ion and spin σ. The
exchange-co ela ion ene gy can be exp essed as he di e ence in kine ic
and in e nal in e ac ion ene gies be ween he in e ac ing many-body sys-
em and he non-in e ac ing auxilia y sys em, whe e elec on-elec on in-
e ac ions a e app oxima ed by EC[n]21.
Once we ha e an exp ession o EKS[n]and unde s and all he componen s
in 2.57 we can de elop he Kohn-Sham equa ions. The Kohn-Sham equa-
ions p o ide a amewo k o compu e he g ound-s a e densi y and ene gy
o he sys em by sol ing an independen pa icle p oblem. These equa ions
a ise om minimizing EKS[n]wi h espec o he densi y n( ,σ)o he e -
ec i e po en ial Vσ
e ( ). The kine ic ene gy TSis exp essed as a unc ional
o he o bi als, while he emaining e ms a e unc ionals o he densi y. By
applying he chain ule, he a ia ional de i a i e o EKS[n]is
∂EKS
∂ψσ∗
i( )=∂TS
∂ψσ∗
i( )+∂Eex
∂n( ,σ)+∂EC
∂n( ,σ)+∂Exc
∂n( ,σ)∂n( ,σ)
∂ψσ∗
i( ). (2.59)
F om Eqs. 2.54 and 2.55, he pa ial de i a i es o he kine ic ene gy and
elec on densi y a e
∂TS
∂ψσ∗
i( )= −1
2∇2ψσ
i( ),∂nσ
∂ψσ∗
i( )=ψσ
i( ). (2.60)
Using he me hod o Lag ange mul iplie s and imposing o hono maliza-
ion cons ain s o he o bi als, ⟨ψσ
i|ψσ′
j⟩=δijδσσ′, we a i e o a Sch ödinge -
like equa ion
−1
2∇2+Vex +VC+Vσ
xcψσ
i( ) = ϵσ
iψσ
i( ), (2.61)
21 Which is, in u n, he Ha ee ene gy, see 3.2 om Re . [66].
2.3 densi y unc ional heo y 35
whe e
Vσ
KS( ) = Vex ( ) + ∂Ec
∂n( ,σ)+∂Exc
∂n( ,σ)
=Vex ( ) + VC( ) + Vσ
xc( ). (2.62)
The Hamil onian can hen be exp essed as
Hσ
KS( )=−1
2∇2+Vσ
KS( ), (2.63)
and he Kohn-Sham equa ions ake he compac o m
[Hσ
KS −ϵσ
i]ψσ
i( ) = 0. (2.64)
The Kohn-Sham equa ions educe he many-body p oblem o an indepen-
den pa icle p oblem whe e he po en ial mus be de e mined sel -consis en ly
wi h he elec on densi y. I he exac exchange-co ela ion unc ional Exc[n]
was known, sol ing hese equa ions would yield he exac g ound-s a e en-
e gy and densi y o he many-body sys em.
Howe e , since Exc[n]encapsula es all many-body co ela ion e ec s, which
scale exponen ially wi h he numbe o elec ons, he exac Exc[n]canno be
de e mined explici ly o sys ems wi h mo e han a ew pa icles [69,70]. In-
deed, his would de ea ou pu pose o downscaling he p oblem’s compu-
a ional complexi y. Ins ead, app oxima ions o Exc[n]a e employed, o m-
ing he ounda ion o mode n-day DFT implemen a ions. We will explo e
hese app oxima ions and hei applica ion in compu a ional me hods in
he ollowing Sec ions.
2.3.3Local Densi y App oxima ion (LDA)
One o he mos simple and in ui i e app oxima ions, bu also one o he
mos ounda ional ones is he LDA, i s in oduced by W. Kohn and L.
J. Sham in 1965 [65]. LDA s ands as LDA app oxima es he exchange-
co ela ion ene gy o an elec on sys em by assuming ha , a e e y poin
in space, he sys em locally esembles an homogeneous elec on gas wi h
densi y n( ). Along his manusc ip , LDA is a ely used, howe e ; we in-
oduce i as he basis ounda ion o mo e de eloped app oxima ions such
as he p esen ed in he ollowing Sec ions.
The exchange-co ela ion ene gy unc ional wi hin LDA is exp essed as
ELDA
xc [n] = Zd ϵhom[n]( )n( ), (2.65)
2.3 densi y unc ional heo y 36
whe e ϵhom[n] ep esen s he exchange-co ela ion ene gy pe pa icle in a
homogeneous elec on gas wi h densi y n( ). We can u he decompose i
in o exchange and co ela ion con ibu ions,
ϵhom
xc [n]( ) = ϵhom
x[n]( ) + ϵhom
c[n]( ), (2.66)
whe e he exchange pa can be exp essed analy ically as
ϵhom
x[n]=−3
43n
π1/3
. (2.67)
Meanwhile, due o i s na u e, he co ela ion pa canno be exp essed ana-
li ically since i a ises om complex many-body non-local elec onic in e -
ac ions ha depend on he ins an aneous posi ions and mo ions o all o he
elec ons. None heless, ϵc[n]can be app oxima ed wi h Mon e Ca lo sim-
ula ions [71] h ough o ms o ϵc ha a e ypically i ed as unc ions o
he a e age elec on dis ance ϵ( s)whe e s=3
4πn1/3. The co esponding
co ela ion po en ial is gi en by
VC( s) = ϵc( s) − s
3
dϵc( s)
d s
. (2.68)
Among he ple ho a o implemen a ions o LDA, p obably he bes well
known one is he one by Pe dew and Zunge [72], which exp esses ϵc( s)
as
ϵPZ
c( s) =
−0.0480 +0.031 ln( s) − 0.0116 s+0.0020 sln( s) s< 1
−0.1423/ 1+1.0529√ s+0.3334 s s> 1.
(2.69)
Due o i s eliance on he p ope ies o a homogeneous elec on gas, LDA
pe o ms well o sys ems whe e elec ons exhibi such beha io , like me -
als and o he bulk solids. Howe e , o inhomogeneous sys ems like a oms
o molecules, whe e he elec on densi y a ies signi ican ly, LDA ends
o be less accu a e. Despi e i s limi a ions, LDA emains ounda ional in
DFT and is widely used due o i s simplici y and compu a ional e iciency,
se ing as a s a ing poin o mo e sophis ica ed app oxima ions.
2.3.4Gene alized G adien App oxima ion (GGA)
Building on he success o LDA, he de elopmen o GGA has lead o an in-
c ease o accu acy an ealibili y on exchange-co ela ion unc ionals. GGA
inco po a es no only he elec on densi y n( )bu also i s g adien ∇n( )
o desc ibe exchange-co ela ion ene gy, allowing i o be e accoun o
inhomogenei ies in he elec on densi y. GGA is a e y eliable app oxima-
ion and i is he app oxima ion o choice o mos o he DFT calcula ions
2.3 densi y unc ional heo y 37
p esen ed in his Thesis due o i s compu a ional cos -accu acy a io.
The i s s ep owa ds GGA was p oposed by Kohn and Sham in hei
seminal pape [65] and la e o malized by He man e al. in Re . [73], lead-
ing o he de elopmen o he G adien Expansion App oxima ion (GEA).
Howe e , GEA did no ep esen a signi ican imp o emen o e he LDA,
as i in oduced se e e issues, such as sum ule iola ion. The GGA was de-
eloped o o e come hese limi a ions by inco po a ing mo e sophis ica ed
unc ionals. We can de ine he GGA unc ional as
EGGA
xc [n] = Zd n( )ϵxc(n( ),∇n( )(2.70)
≡Zd n( )ϵhom
x(n)Fxc(n,∇n), (2.71)
whe e Fxc(n,∇n)is a dimensionless enhancemen ac o ha depends only
on nand i s g adien , while ϵhom
xc (n)is he exchange-co ela ion ene gy pe
pa icle o an homogeneous elec on gas as happening in LDA.
Among he nume ous GGA implemen a ions, he mos widely used im-
plemen a ion o GGA (aswell as he mos used in he wo k p esen ed
in his manusc ip ) is he one by Pe dew Bu ke and E ze ho (PBE) [74].
PBE was p esen ed as a GGA imp o ing he p e iously exis ing Pe dew-
Wang 1991 (PW91) [75] GGA, which was deemed as non- anspa en , o e -
pa ame ized and ailing in se e al si ua ions. PBE is cons uc ed based
on a sy ema ic app oach ha imp o es upon LDA by using exac physical
cons ain s a he han i ing o expe imen al da a like pu ely empi ical
unc ionals. Thanks o ha , his GGA is mo e ans e able ac oss di e en
sys ems and sa is ies exac cons ain s imposed upon i as: co ec asymp-
o ic beha iou o Ex, co ec scaling p ope ies a low and high densi y
egimes and complies wi h he Coulomb’s ene gy lowe bound p oposed
by Lieb-Ox o d [76], p e en ing unphysical Exbeha iou and Exchole no -
maliza ion.
The exchange pa o he PBE unc ional modi ies he enhancemen ac-
o Fxc in 2.70, ha we will deno e as Fx, using he dimensionless educed
g adien s, de ined as
s=|∇n( )|
2kF( )n( ), (2.72)
whe e kF( )is he Fe mi wa e ec o 22 and i s ela ed o n( )by
n( ) = kF( )3/(3π)2. (2.73)
22 The Fe mi wa e ec o de ines he adius o he Fe mi sphe e in momen um space. All
elec on s a es wi h wa e ec o k<kFa e occupied a ze o empe a u e while s a es
wi h k > kFa e unoccupied. In he homogeneous ee elec on gas limi he Fe mi sphe e
coincides wi h he Fe mi su ace.
2.3 densi y unc ional heo y 38
F om 2.70,EGGA
xis exp essed as
EGGA
x[n] = Zd FPBE
x(s)n( )ϵhom
x[n], (2.74)
whe e
FPBE
x=1+κ 1−1
1+µs2
κ!, (2.75)
wi h κ=0.804 and µ=0.21951. Meanwhile, he co ela ion pa can be
exp esed as a co ec ion o LDA co ela ion as ollows,
EGGA
c[n] = Zd n( )hϵhom
c[n] + H( s, )i, (2.76)
whe e is a dimensionless g adien and sis he a e age in e -elec on
dis ance and can be exp essed as
≡∇n( )
2ksn( ), wi h ks=4kF
π2
. (2.77)
Meanwhile, H( s, )encodes he co ec ion o e LDA; in he case o PBE i
is
HPBE( s, ) = β2
2α ln1+2α
β
2+A 4
1+A 2+A2 4, (2.78)
wi h
A=2α
βexp−2αVhom
c[n]
β2−1
,
α=0.0716,β=0.066725,
=|∇n( )|
2ksn( ),ks=4kF
π1/2
.
Like LDA, GGA is mos accu a e when exchange-co ela ion ene gies a e
ela i ely small, such as in weakly co ela ed sys ems. This makes he GGA
pa icula ly sui able o s udying sys ems like bulk me als, semiconduc o s,
and simple molecules. Fo he sys ems explo ed in his manusc ip , he
GGA (and mos conc e ely PBE) p o ides a eliable balance o compu a-
ional e iciency and accu acy, making i he me hod o choice o exchange-
co ela ion app oxima ions.
PBE is gene ally chosen along his manusc ip due o i s balance be ween
2.3 densi y unc ional heo y 39
accu acy and compu a ional cos . I pe o ms well o bulk ma e ials, su -
aces and in e aces and has a ai ly accu a e he mochemical and s uc-
u al p edic ions (we also use i o elaxing s uc u es). I has a decen
band gap es ima ion when compa ed o LDA, howe e i ends o unde -
s ima e band gaps due o i s sel -in e ac ion e o ; ha is why, along his
manusc ip we use he modi ied Becke-Johnson (mBJ) exchange po en ial
[77,78] when ying o de e mine exac band gaps in Chap e 3. Mo eo e ,
beyond i s applicabili y, PBE se es as he ounda ion o many imp o ed
unc ionals, like PBEsol [79], op imized o solids and densely packed sys-
ems; HSE06 [80], which is an hyb id unc ional ha co ec s PBE’s band
gap unde s ima ion and dW-DF [81], co ec ing i s de iciencies in weak
in e ac ions by adding an de Waals co ec ions.
2.3.5A b ie no e on modi ied Becke-Johnson exchange po en ial
As al eady men ioned in he p e ious Sec ion, in Chap e 3we use mBJ
o de e mine a mo e accu a e bandgap. mBJ is no a GGA in he con en-
ional sense, i is a semilocal exchange po en ial ha goes beyond GGA bu
does no uly each he le el o hyb id un ionals like he p e iously men-
ioned HSE06 o nonlocal exchange me hods like GW app oxima ion [82–
85] (used o ob ain sel -ene gies in many-body sys ems). I is so because
i uses a dependence on he kine ic ene gy densi y, ( ), and a sys em-
dependen mixing pa ame e C ha imp o es he ea men o Ex,
EmBJ
x,σ( ) = cEBR
x,σ+ (3c −2)1
πs5
6
σ( )
nσ( ), (2.79)
whe e σ=1
2PNσ
i=1ψ∗
i,σψi,σis he kine ic-ene gy densi y and
BR
x,σ( )=− 1
bσ( )1−e−xσ( )−1
2xσ( )e−xσ( ), (2.80)
is he Becke-Roussel po en ial [86], p oposed o model he Coulomb po en-
ial c ea ed by he exchange hole. xσis de e mined om an equa ion in ol -
ing nσ,∇nσ,∇2nσ, and σ. Then bσis calcula ed wi h bσ= [x3
σe−xσ/8πnσ]1/3.
cis a pa ame e chosen o depend linea ly on he squa e oo o he a e age
o ∇n( ′)
n( ′),
c=α+β1
Vcell Zcell
d ∇n( ′)
n( ′), (2.81)
whe e αand βa e wo ee pa ame e s and Vcell is he uni cell olume.
This spa ially dependen exchange po en ial allows mBJ o cap u e e ec s
simila o hose o GW app oxima ion.
2.3 densi y unc ional heo y 40
The mBJ po en ial e ec i ely enhances he exchange po en ial in egions
o low elec on densi y and high densi y g adien s. This is c ucial because
he unde es ima ion o band gaps in LDA and GGA is la gely due o an in-
co ec desc ip ion o he sc eened exchange po en ial. Mo eo e , he mBJ
co ec ion inc eases he ene gy sepa a ion be ween occupied and unoccu-
pied s a es, he eby imp o ing he band gap p edic ion. This beha io is
pa icula ly ele an in semiconduc o s and insula o s, whe e he conduc-
ion band minimum is o en oo low in LDA and GGA, while he alence
band maximum is sligh ly oo high. The mBJ po en ial app op ia ely shi s
hese bands, leading o be e ag eemen wi h expe imen al band gaps. Be-
yond his, unlike hyb id unc ionals and he GW app oxima ion, which
equi e compu a ionally expensi e nonlocal exchange e ms, he mBJ po-
en ial emains a semilocal unc ional while inco po a ing empi ical i ing
pa ame e s. This allows i o closely ma ch expe imen al da a wi hou a
signi ican compu a ional bu den.
In conclusion, he mBJ exchange po en ial signi ican ly imp o es band gap
p edic ions in semiconduc o s and insula o s by in oducing a nonlocal
exchange-like co ec ion h ough a kine ic-ene gy-dependen e m. I o e s
supe io accu acy compa ed o LDA and GGA and pe o ms compe i i ely
wi h hyb id unc ionals while main aining a ela i ely low compu a ional
cos . This balance be ween accu acy and e iciency is he p ima y eason
we employ i o band accu a e band gap calcula ions. None heless, o
gene al pu poses, GGA+PBE is mo e han enough o an accu a e sys em-
a ic dec ip ion o he elec onic s uc u es o he sys ems unde s udy o
his Thesis.
2.3.6Concluding ema k
Un il now, we ha e de eloped a amewo k o simula ing he elec onic
s uc u e o ma e by employing he Hohenbe g-Kohn heo ems o de i e
he Kohn-Sham equa ions, as well as in oducing app oxima ions o he
exchange-co ela ion ene gy in hese equa ions. Howe e , a p ac ical im-
plemen a ion o sol ing he Kohn-Sham equa ions would s ill equi e he
explici ea men o all elec ons (bo h co e and alence) in he sys em
unde s udy, which can be compu a ionally p ohibi i e o he sys ems
conside ed in his Thesis. In he ollowing Sec ion, we will explo e how
o o e come his challenge by eplacing he complex in e ac ions o igh ly
bound co e elec ons wi h an e ec i e po en ial, enabling a mo e e icien
calcula ion o alence elec onic s a es, key o ou eseach.
2.5 bloch heo em and plane-wa ebasis 47
A ansla ion ope a o TRcan hen be de ined o shi he sys em by a la ice
ec o Ras
TR ( ) = ( +R). (2.98)
A c ys alline solid is in a ian unde he ac ion o any p imi i e ansla ion
ope a o Taio hei combina ions26. Because hese ansla ion ope a o s
commu e wi h he Hamil onian, he Hamil onian eigens a es can be also
labeled by he eigen alues o he ansla ion ope a o . These eigen alues
ake he o m eik·R, whe e kis he c ys al momen um, a iple o phases.
The implica ions o his e ec i e pe iodic ea men a e cap u ed in he
Bloch’s heo em,
Theo em 2.5.1The solu ions o he Sch ödinge equa ion in a pe iodic po en-
ial can be exp essed as plane wa es modula ed by pe iodic unc ions u( ):
ψ( ) = eik· u( ), (2.99)
whe e u( )is a unc ion pe iodic wi h he la ice.
When applied o ou Kohn-Sham equa ions, we can w i e he Kohn-Sham
s a es ϕias ϕσ
n,kwhe e iis now decoupled in o k, associa ed o a pa ic-
ula plane-wa eeik· and n, associa ed o a pa icula pe iodic modula ion
un,k( ),
ϕnk( ) = eik· un,k( ), (2.100)
whe e kis de ined in he i s B illouin zone (IBZ) and n is called band
index. Since any pe iodic unc ion can be expanded in a comple e se o
Fou ie componen s, we can exp ess he pe iodic unc ion un,k( )in e ms
o ecip ocal la ice ec o s G
un,k( ) = X
G
cn,k+Gei(G)· (2.101)
whe e he plane-wa eexpansion na u ally leads o a Fou ie ep esen a ion
o he Kohn-Sham eigen unc ions,
ϕnk=X
G
cn,k+Gei(k+G)· . (2.102)
In o de o p e en ene gy di e gences om he in ini e numbe o Gcom-
ponen s, we in oduce an ene gy cu o Ecu , disca ding all componen s
wi h
Ecu ⩽1
2|k+G|2. (2.103)
26 Since ansla ion ope a o s commu e.
2.5 bloch heo em and plane-wa ebasis 48
This unca ion educes he compu a ional cos and ensu es he con e -
gence o calcula ions, usually is e e ed in a DFT en i onmen as he kine ic
cu o . Wi h his app oach, he Kohn-Sham equa ions become
X
G′
HG,G′(k)cn,k+G′=ϵi(k)cn,k+G, (2.104)
whe e he Hamil onian ma ix elemen s a e
HG,G′(K)≡ ⟨k+G|HKS|k+G′⟩=1
2|k+G|2δG,G′+VKS(G−G′).
(2.105)
In Eq. 2.104,ihas been spli in o IBZ wa e- ec o kand band index n,
which allows us o cons uc bands uc u es om sol ing his Hamil o-
nian, by labelling each s a e by he wa e ec o k. This c ys al momen um
is conse ed analogously o o dina y momen um in ee space modulo
he addi ion o any ecip ocal la ice ec o Gand hence, solu ions o he
Hamil onian a e pe iodic in kand unique when exp essed in he IBZ o
he ecip ocal la ice, allowing bands o build a comple e desc ip ion o he
elec onic beha iou o he sys em. Fu h e mo e, he kdesc ip ion allows
us o in eg a e p ope ies such as he numbe o elec ons, he o al ene gy
e c. Fo example, o a unc ion n(k), whe e ndeno es he disc e e band
index, he a e age alue is
¯
n=1
NkX
k
n(k)→Ωcell
(2π)3ZBZ
dk n(k), (2.106)
whe e Ωcell is he olume o a p imi i e cell in eal space. The DOS pe
uni cell can be exp essed as
N(ϵ) = 1
NkX
nX
k
δ(ϵ−ϵn,k)
=Ωcell
(2π)dZBZ
dkδ(ϵ−ϵn,k). (2.107)
As DOS, any quan i y can be in eg a ed o e he IBZ, like op ical esponse
enso s27. This Sec ion is one o he mos impo an o ou Thesis, hanks
o i we can oba in band s uc u es, DOS and a ple ho a o esul s om
ou DFT calcula ions. In he ollowing, we will ennume a e all he ways in
which DFT can educe compu a ional cos .
27 The e is a ca ea o his: e en hough he plane-wa e basis amewo k allows o calcula ion
o i s and second o de dielec ic esponses, usually said calcula ions a e no pe o med
in a plane-wa e basis since he k−g id is a he spa se due o compu a ional limi a ions.
Usually, said calcula ions a e pe o med in a Wannie basis ha allow o dense meshes
wi hou a hea y compu a ional cos . We go h ough his me hod in Sec. 2.7.
2.6 how d educes compu a ional cos 49
2.6 how d educes compu a ional cos
In he p e ious Sec ions o his Chap e we ha e unde s ood why and
how DFT educes he compu a ional complexi y o sol ing he many-body
Sch ödinge equa ion by se e al in e ela ed mechanisms. In he ollowing,
we will ennume a e and go h ough hem as a o m o summa y:
1.Reduc ion o a single-pa icle amewo k wi h he Kohn-Sham ap-
p oach: The Kohn-Sham o mula ion o DFT eplaces he ull many-
body in e ac ing elec on p oblem wi h a sys em o non-in e ac ing
elec ons mo ing in an e ec i e po en ial, Ve ( ). This e ec i e po en-
ial encapsula es he e ec s o elec on-elec on in e ac ions h ough
he exchange-co ela ion ene gy unc ional, Exc[n], which depends
only on he elec on densi y n( ). And, as a esul :
• The compu a ional cos will scale polynomially wi h he numbe
o elec ons N, ypically O(N3); ins ead o exponen ially.
• Sol ing he Kohn-Sham equa ions in ol e i e a i e sel -consis en
ield me hods, which a e easible compu a ionally o sys ems
wi h hund eds o housands o a oms.
2.Dimensionali y educ ion: Th ough he Hohnenbe g-Kohn heo ems,
we can es ablish ha all p ope ies o he elec onic sys ems can be
de e mined uniquely by he g ound-s a e elec on densi y, which is
a3D scala unc ion, independen ly o he numbe o elec ons in
he sys em. This will d ama icly educe he dimensi onali y om he
3N-dimensional wa e unc ion in o a 3D densi y ha enables ac able
calcula ions o sys ems wi h many elec ons.
3.Using app oxima e unc ionals: The exac exchange-co ela ion unc-
ionals a e no known, bu using p ac ical app oxima ions like LDA o
GGA p o ide a easonable accu acy o a wide ange o sys ems while
emaining compu a ionally e icien . Wi h hese app oxima ions DFT
a oids he di ec calcula ion o co ela ed elec on wa e unc ions and
eplace he complex elec on-elec on in e ac ions wi h manageable
analy ically o semi-empi ical exp essions.
4.Explo ing ansla ional symme y: In pe iodic sys ems, DFT uses he
Bloch’s heo em o ep esen elec onic s a es as Bloch unc ions, e-
ducing he p oblem o a single uni cell and sampling in ecip ocal
space. By using ansla ional symme y we educe he size o he o-
diagonalize-Hamil onian and limi all ou compu a ions o he IBZ.
5.Basis se simpli ica ions: By using plane wa es o localized basis
se s, combining wi h ene gy cu o s, we allow he unca ion o high-
ene gy componen s in he solu ion, d as ically educing he numbe
2.7 wannie unc ions 50
o basis unc ions needed. This unca ion is a middle-g ound be-
ween accu acy and compu a ional e iciency.
6.Reducing he numbe o elec ons by elimina ing co e elec ons
wi h Pseudopo en ials: Pseudopo en ials eplace he explici ea -
men o he bound co e elec ons wi h an e ec i epo en ial ac ing
only on alence elec ons. By his app oach we educe he numbe
o wa e unc ions o be calcula ed and simpli y ehe po en ial land-
scape 28, imp o ing nume ical s abili y and educing he basis se
size eque imen s.
7.E icien algo i hms and pa alleliza ion: Mode DFT implemen a-
ions make use o a ple ho a o ad anced nume ical me hods due
o i s na u e. Be ween hese me hods we ha e spa se ma ix me hods
o la ge sys ems, i e a i e diagonaliza ion schemes and e icien as
ou ie ans o ms o plane-wa eexpansions. These me hods allow
DFT o scale e icien ly o la ge and complex sys ems.
By ans o ming he many-body p oblem in o a se ies o single-pa icle
equa ions and employing app oxima ions, DFT achie es an excep ional
balance be ween accu acy and compu a ional cos . While some challenges
emain, such as he exac ea men o exchange-co ela ion e ec s and
s ongly co ela ed sys ems, DFT’s e iciency and lexibili y ha e made i
he p ima y ool o in es iga ing he elec onic s uc u e o ma e . The
amewo k laid ou in his Chap e o ms he ounda ion o he me hods
and app oxima ions explo ed h oughou his Thesis.
2.7 wannie unc ions
In he s udy o pe iodic sys ems, Bloch’s heo em p o ides a con enien
ep esen a ion o he elec onic s a es as ex ended plane wa es modula ed
by pe iodic unc ions. Howe e , he delocalized na u e o Bloch unc ions
blu s a possible eal-space in e p e a ion o elec onic p ope ies. To his
end, Wannie unc ions, in oduced by G. Wannie in 1937 [15], p o ide a
localized eal space basis ha is o hono mal and spans he same subspace
as he Bloch eigens a es o one o se e al bands. These unc ions a e pa ic-
ula ly ad an ageous o cons uc ing simpli ied models, analyzing chemi-
cal bonding and s uding anspo phenomena. Mo eo e , by ocusing on
speci ic se s o bands, Wannie unc ion can acili a e he educ ion in com-
pu a ional complexi y o elec onic s uc u e p oblems, such as he calcu-
la ion o op ical esponse enso s as in Sec. 3.2.5and a ple ho a o o he
28 The e ec i e po en ial eplacing co e elec ons is much smoo he when compa ed o he
ac ual Coulomn po en ial nea he nucleus since i does no accoun o he high- equency
oscilla ions o co e elec on.
2.7 wannie unc ions 51
p ope ies. In he ollowing, we in oduce he concep o Wannie unc-
ions and hei applica ions, bo h in ou esea ch and in he b oade ield,
highligh ing hei c ucial ole in compu a ional condensed ma e physics.
F om he Bloch heo em, we can w i e he wa e unc ions o a pe iodic
Hamil onian Has
ψn,k( ) = eik· un,k( ). (2.108)
These Bloch unc ions a e only de ined up o a phase ac o , allowing o a
gauge an o ma ion o he o m
ψn,k( )→˜
ψn,k=eiθn(k)ψn,k( ), (2.109)
whe e θn(k)is a di e en iable phase unc ion. The Wannie unc ion o
band n, associa ed wi h he la ice ec o Rm, is de ined as he Fou ie
ans o m o Bloch s a es o e he B illouin zone,
ωn( −Rm) = Ωcell
(2π)3ZBZ
dke−ik·Rmψn,k( ), (2.110)
whe e Ωcell is he uni cell olume and hese unc ions decay exponen ially
as | −Rm|→∞, ensu ing hei localiza ion. The o iginal Bloch unc ions
can be econs uc ed om he Wannie unc ions ia he in e se ans o m,
ψn,k( ) = X
m
eik·Rmωn( −Rm)(2.111)
2.7.1Gene alized Wannie Func ions o Mul iple Bands
Wannie unc ions can also be cons uc ed o a g oup o bands by o ming
linea combina ions o Bloch s a es. Fo Nbands, he pe iodic pa o he
Bloch unc ion, un,k, can be o a ed using a uni a y ma ix U(k),
uW
i,k=X
n
Un,i(k)un,k. (2.112)
The esul ing Wannie unc ions a e gi en by
ωi( −Tm) = Ωcell
(2π)3ZBZ
dkeik·( −Tm)X
n
Un,i(k)un,k( )(2.113)
These gene alized Wannie unc ions e ain o hono mali y and p o ide a
lexible basis o cap u ing elec onic p ope ies.
2.7 wannie unc ions 52
2.7.2Non-Uniqueness and Maximally localized Wannie unc ions
Due o he gauge eedom in choosing θn(k), he Wannie unc ions a e no
uniquely de ined. Indeed, he choice o gauge can signi ican ly in luence
hei shape and localiza ion. In o de o add ess his, we can cons uc
maximally localized Wannie unc ions (MLWFs) by minimizing he sp ead
unc ional
Ω=X
ih⟨ 2⟩i−⟨ ⟩2
ii, (2.114)
whe e ⟨ ⟩iand ⟨ 2⟩ia e he expec a ion alues o posi ion ope a o s o he
i- h Wannie unc ion. This minimiza ion ensu es a consis en and physi-
cally meaning ul ep esen a ion o localized s a es. The pu sui o MLWFs
add esses he a bi a iness in gauge choice while o e ing he addi ional
bene i o simpli ying calcula ions. Thei localiza ion educes he o e lap
be ween neighbo ing unc ions, esul ing in spa se and mo e compu a-
ionally e icien Hamil onians. MLWFs a e pa icula ly well-sui ed o con-
s uc ing igh -binding models, calcula ing Be y phases, and de e mining
pola iza ion. By using maximally localized Wannie unc ions, spu ious
oscilla ions and long- ange in e ac ions a e minimized, enabling p ecise
eal-space analyses and p o iding a clea e physical unde s anding. This is
because MLWFs yield well-de ined cen e s o cha ge, enhancing hei in e -
p e abili y and making hem a aluable ool o explo ing elec onic s uc-
u e. Fo example, he code employed in Ch. 3;Wannie 90 [98], is based on
inding MLWFs.
2.7.3Applica ions and Wannie in e pola ion
Wannie unc ions a e key in cons uc ing igh -binding models, calcula -
ing anspo p ope ies and s udying opological in a ian s. Addi ionally,
hei localiza ion enables Wannie in e pola ion, allowing elec onic p op-
e ies o be compu ed on ine k−space g ids wi hou he need o ull
diagonaliza ion o he Hamil onian. The wo k low is as ollows:
1.Ini ial DFT calcula ion: Kohn-Sham equa inos a e sol ed in a coa se
k-poin g id 29 in o de o ob ain eigen alues and eigen unc ions a
disc e e poin in he ecip ocal space.
2.Cons uc ion o Wannie Func ions: Bloch wa e unc ions a e ans-
o med in o Wannie unc ions using uni a y an o ma ions. A e
ha , he Wannie unc ions a e maximally localized by minimizing
he sp ead unci onal.
29 Leading hese calcual ions o be compu a ionally inexpensi e.
2.7 wannie unc ions 53
3.In e pola ion o Hamil onian in ecip ocal space: In ecip ocal space,
he elec onic Hamil onian is ep esen ed in Wannie basis as
Hmn(k) = X
R
eik·R⟨ωm( )|H|ωm( +R)⟩, (2.115)
whe e ⟨ωm( )|H|ωm( +R)⟩decay apidly due o MLWFs being spa-
ially localized. This allows he Hamil onain o be e icien ly in e po-
la ed om a coa se g id o an a bi a y k-poin g id in he BZ 30
4.Applica ions: This in e pola ed Hamil onian can be diagonalized on
an a bi a y se o k-poin s wi hou adi ional ab-ini io calcula ions.
This has a ple ho a o applica ions such as high- esolu ion bands uc-
u es, Fe mi su aces, DOS, Be y-phase p ope ies and op ical e-
sponses as in Sec. 3.2.5. Ano he g ea applica ion o Wannie unc-
ions is ha hey allow he sys ema ic cons uc ion o igh -binding
models on a localized basis ha accu a ely ep oduce he band s uc-
u e o he ma e ial in a minimal basis se .
2.7.4Why and when using Wannie in e pola ion
Wannie in e pola ion me hod and i s compu a ional applica ion h ough
Wannie 90 [98] is o pa amoun impo ance in mode n compu a ional con-
densed ma e esea ch. I builds upon DFT as bo h a heo e ical amewo k
and a p ac ical ool, enabling he calcula ion o a wide ange o p ope -
ies on dense, high- esolu ion k-g ids. Fo ins ance, in Ch. 3, we compu e
he op ical conduc i i y o (TaSe4)3I using a 91 ×91 ×71 k-mesh, whe eas
he unde lying DFT calcula ions a e pe o med on a signi ican ly coa se
11 ×11 ×5 k-mesh. Di ec ly compu ing he op ical conduc i i y on such a
dense g id using con en ional DFT would be compu a ionally p ohibi i e.
To illus a e he e iciency gain, conside a naï e es ima ion: assuming an
ideal pa alleliza ion scheme wi h no o e head and he same kine ic en-
e gy cu o (i.e., he same basis se ) o bo h cases, le us assume ha he
sel -consis en DFT calcula ion on he 11 ×11 ×5g id akes 1hou o com-
ple e. Gi en he polynomial scaling wi h Nkand an assumed sublinea
scaling e iciency o 80% due o pa alleliza ion, a sel -consis en calcula ion
on a 91 ×91 ×71 k-g id would ake app oxima ely 3days, dis ega ding
p ac ical limi a ions such as RAM sho ages o pa alelliza ion o e head. In
con as , in ou expe ience using Wannie in e pola ion, he op ical con-
duc i i y calcula ion equi ed only 1.5 imes he compu a ional ime o he
sel -consis en DFT calcula ion, leading o an o e all speed-up o a ac o
o 50. Mo eo e , his es ima e conside s only he sel -consis en s ep; com-
pu ing op ical conduc i i y also equi es e alua ing he de i a i es o he
o bi als wi h espec o k, which in a di ec DFT app oach would equi e
30 This p ocess can be u he ly op imized using algo i hms as he as Fou ie T ans o m.
2.7 wannie unc ions 54
an addi ional compu a ionally expensi e s ep, u he inc easing he o al
un ime.
E en hough he me hods p esen ed in his Sec ion o e immense com-
pu a ional powe and signi ican compu a ional cos educ ion, i is impo -
an o discuss why, o example, we do no use hem o model cha ge
ans e in Chap e 4. The p ima y challenge when wo king wi h Wannie
unc ions lies in he app op ia e choice o basis. The selec ion o a basis in
Wannie unc ion calcula ions is a use -p o ided inpu ha equi es bo h
expe ise in he physical sys em unde s udy and a deep unde s anding
o he compu a ional implemen a ion31. A poo ly chosen basis, combined
wi h o e - i ing h ough a long localiza ion p ocess, can lead o a model
ha lacks meaning ul physical insigh , ul ima ely esul ing in inco ec p e-
dic ions.
Fo ins ance, he p ocess o maximal localiza ion i sel can b eak c ucial
c ys al symme ies, which a e essen ial o unde s anding phenomena such
as op ical esponses o Be y phase p ope ies32. Assessing whe he a Wan-
nie iza ion is well-execu ed is highly non i ial and equi es deep knowl-
edge o bo h he sys em unde s udy and he Wannie iza ion p ocess i -
sel . A pa icula example o he issues ha can a ise when using MLWF
a e he p oblems o in e p e ing ma e ial p ope ies h ough Wannie -
in e pola ed igh -binding models. While hese models a e buil using lo-
calized Wannie unc ions (which o en o igina e om ini ial ial unc ions
ha a e o bi al-like), any o bi al- esol ed o symme y-dependen in o ma-
ion abou he econs uc ed bands may become obscu ed due o he disen-
angling and Wannie iza ion p ocedu es, pa icula ly when maximal local-
iza ion is applied. In many cases, o p ese e o bi al cha ac e and symme-
y p ope ies, a one-sho Wannie iza ion app oach is ecommended (see
Sec. II.I.1o [99]).
Despi e hese nuances, he po en ial pi alls o Wannie iza ion should no
be seen as p ohibi i e bu a he as aspec s o be ca e ully managed. A
well-chosen basis ha espec s he symme ies o he sys em, o he appli-
ca ion o symme y-cons ained maximal localiza ion, can mi iga e many
o he p e iously men ioned issues. Fu he mo e, in some cases, he appa -
en b eaking o symme y may i sel e eal hidden physical phenomena,
making Wannie iza ion an impo an diagnos ic ool in condensed ma e
physics. In conclusion, Wannie in e pola ion and, mo e speci ically ML-
31 In his ex , we conside only Wannie 90 [98].
32 The gauge choice in Wannie iza ion may no always p ese e he ull symme y o he
o iginal Bloch wa e unc ions. Fo example, i a ma e ial exhibi s in e sion symme y in
i s Bloch s a es bu he localiza ion p ocedu e selec s an asymme ic Wannie gauge, he
inal Wannie unc ions may no espec in e sion symme y. This can signi ican ly a ec
Be y phase calcula ions, elec onic pola iza ion, and o he opological p ope ies.
2.7 wannie unc ions 55
WFs, cons i u es a powe ul ool o condensed ma e esea ch. Howe e ,
he de elopmen o sys ema ic amewo ks based on MLWFs emains a
challenge due o he equi ed ca e ul selec ion o basis unc ions. To ad-
d ess his, e o s ha e been made owa d au oma ic Wannie iza ion p oce-
du es [100], hough we conside hem o s ill be unde de elopmen .
Pa III
RESULTS
3.1 in oduc ion 63
hose ound in CDWs. Mo eo e , unde s anding CDW ansi ions is c ucial
o in e p e ing cha ge anspo measu emen s and explo ing uncon en-
ional o de ing phenomena in laye ed ma e ials. In la e Sec ions, we will
e isi hese concep s in he con ex o speci ic ma e ials and hei in e play
wi h o he collec i e quan um s a es.
3.1.4TMCs (MSe4)nIwi h M=Nb,Ta.
The se ies o TMCs (MSe4)nIwi h M=Nb,Ta; a e made o weakly cou-
pled MSe4chains, each o hem hos ing a dz2-de i ed 1D band wi h ac-
ional illing δ= (n−1)/2n. Repo ed s uc u es occu a n=2,3,10
3i.e.
δ=0.25,0.33,0.35 espec i ely. These ac ionally illed quasi-1D bands a e
p one o CDW ins abili ies which ha e been ex ensi ely s udied [17,18,
101,103,104,111–119]. Among he Nb-based compounds, he n=2and
n=10
3 a ian s exhibi clea incommensu a e CDW ansi ions o igina -
ing om a me allic pa en s a e. These ansi ions display well-es ablished
CDW phenomenology, including non-linea anspo due o CDW depin-
ning, e lec ing he s ong coupling be ween he elec onic and la ice de-
g ees o eedom. In con as , he n=3compound, (NbSe4)3I, s ands ou
as he only membe o he se ies ha does no exhibi a con en ional CDW.
Ins ead, i unde goes a s uc u al ansi ion and shows ac i a ed semicon-
duc ing anspo . A key o unde s anding his beha io lies in i s unusual
elec onic con igu a ion: he ma e ial ea u es a mixed- alence s a e among
he Nb a oms. Assuming a simpli ied ionic model wi h Se2−and I−, he
a e age Nb oxida ion s a e is app oxima ely +4.33, implying a pe iodic co-
exis ence o Nb4+(4d1) and Nb5+(4d0) ions. This leads o a si e-selec i e
dis ibu ion o d-elec ons along he quasi-one-dimensional chains o ace-
sha ing NbSe6oc ahed a. While he Nb4+si es con ibu e localized d1elec-
ons p one o Peie ls ins abili ies and gap o ma ion, he Nb5+si es ac
as elec onically ine space s. The esul ing pe iodic modula ion o he
elec onic densi y mimics he e ec s o a Peie ls dis o ion, bu wi hou e-
qui ing a pu ely elec onic ins abili y, as p esen ed in he in oduc ion o
his Chap e . Despi e his, se e al puzzles ha e p ecluded a consis en un-
de s anding o he n=3compound wi hin simple band heo y: mul iple
s uc u al ansi ions ha e been epo ed [103,104,114,115], anspo gaps
a y signi ican ly ac oss s udies [18,112,120–122], and ARPES and op ical
conduc i i y expe imen s [123,124] e eal a gap much la ge han hose
obse ed in anspo , sugges ing he p esence o hidden elec onic s uc-
u e and low-spec al-weigh s a es no cap u ed by con en ional models.
Fo he isoelec onic Ta compounds, only n=2,3 a ian s a e epo ed
in he li e a u e [16,120,124]. While (TaSe4)2I has seen a enewed in e es
[33] in he con ex o axionic CDW in Weyl semime als [34,35], he knowl-
edge abou (TaSe4)3I is a he sca ce, as i was assumed o beha e mos ly
3.1 in oduc ion 64
like i s Nb coun e pa [16,120,124]. In a ecen de elopmen , howe e , a
poly ype o (TaSe4)3I has been ound o be me allic a oom empe a u e,
wi h a e omagne ic ansi ion a 8K, and supe conduc i i y coexis ing
wi h e omagne ism a Tc=3K [36]. This coexis ence is unusual on i s
own [37,38], bu i is all he mo e su p ising gi ing he isoelec onic Nb
compound semiconduc ing beha iou .
The con lic ing anspo and ARPES esul s o (NbSe4)3I, along wi h he
su p ising low empe a u e beha io o (TaSe4)3I e eal ha an unde s and-
ing o hese ma e ials in e ms o a band s uc u e pic u e we e lacking.
Along he nex pages o his Sec ion we de elop a heo e ical and compu-
a ional amewo k o unde s anding he p e ious li e a u e and u he
de elop his unde s anding expe imen ally. Fi s ly, we combine ab-ini io
calcula ions o all he known s uc u es o bo h compounds along a igh -
binding analysis o explain many o he elec onic ea u es o hese sys ems.
All phases a e ound o be semiconduc ing, wi h gaps ha di ec ly co e-
la e wi h he amoun Nb o Ta ime iza ion dis o ion6. We discuss how
hese gaps compa e wi h he anspo obse a ions and gene ally ind
good ag eemen . Howe e , we also show ha while he gap is o mally
di ec and loca ed a Γ, he e is an app oxima e in-plane ansla ional sym-
me y ha leads o negligible spec al weigh o he alence band edge, so
ARPES and op ical conduc i i y ac ually p obe highe bands and lead o a
misiden i ica ion o he ue gap. To conclude, we iden i y a spin-spli Van
Ho e singula i y in he alence band edge in he new (TaSe4)3I poly ype
which c osses he Fe mi le el o a e y small amoun o hole doping, and
we a gue his may p o ide a consis en explana ion o e omagne ism and
po en ially luc ua ion media ed iple pai ing.
3.1.5Phenomenology o e iew on (MSe4)3I(M=Nb,Ta) TMCs.
Fo he n=3compounds, (NbSe4)3I is he bes s udied one. I s high em-
pe a u e phase has a e agonal, cen osymme ic c ys al s uc u e wi h
space g oup (SG) P4/mnc (No. 128, poin g oup D4h). As empe a u e is
lowe ed, i unde goes a s uc u al phase ansi ion a Tc1 ∼274 −280 K wi h
he esul ing space g oup P¯
421c(No. 114, poin g oup D2d). A second s uc-
u al ansi ion o a phase wi h SG P¯
4(No. 81, poin g oup S4) a Tc2 ∼90
K has also been epo ed [101,104,114,115], bu no in all expe imen s. In
his wo k, we will e e o hese s uc u es only by hei poin g oup. The
pa e n o symme y b eaking is hus D4h →D2d →S4.
Resis i i y measu emen s a e gene ally consis en wi h semiconduc ing be-
ha io wi h ac i a ed esis i i y ρ∝eEg/2kBTand a gap Eg ha appea s o
6See Eq. 3.5.
3.1 in oduc ion 65
change h ough he phase ansi ions. Abo e Tc1, alues o Eg=190-220
meV [16,119] we e epo ed. A esis i i y kink was always obse ed a
Tc1, and below i wo ypes o samples we e epo ed o exis [16], ini ially
indis inguishable by s uc u al measu emen s [16,112]. In ype I samples
Egis educed o alues in he ange 20-70 meV [16,18,112,121] all he
way o lowes empe a u es measu ed. In ype II samples, a b oade kink
is obse ed which leads o low empe a u e gaps o 110-130 meV [16,119].
Howe e , in some samples ∂lnρ/∂T−1ne e la ens o a cons an Eg alue,
bu a he con inuously dec eases a e a maximum [112,122], challenging
he iew o s anda d ac i a ed anspo . Di e ing obse a ions also in-
clude a epo o Eg=97 meV o T > Tc1 and 222 meV o T << Tc1 [115],
o e en a an ab up change a 180 K om 345 meV o 22 meV [113]. Ano he
cha ac e is ic ea u e o ype II samples is ha swi ching o a s a e o lowe
esis ance can be induced a high cu en s and low empe a u es [122]. This
beha io disappea s a 140 K, and is comple ely absen in ype I c ys als.
A pa ial solu ion o hese anspo puzzles was o e ed in Re . [119] which
epo ed ha he second s uc u al ansi ion was only obse ed in samples
assigned o ype II in anspo . The o e all sugges ed pic u e would hen
be ha all h ee s uc u es a e semiconduc o s: he ini ial gap o 190-220
meV o he D4h s uc u e is educed o 20-70 he D2d s uc u e, and o
samples wi h he second ansi ion i inc eases again in he S4s uc u e o
110-130. While ou lie s o his pic u e do exis , we conside his o be he
a e age beha iou o compa e wi h ou calcula ions. Ea ly ARPES expe -
imen s a 300 K also a emp ed o de e mine he spec al gap [123,125].
The alence band maximum was ound o be a 750 meV below EF, p o id-
ing a lowe bound on he gap which is much la ge han he one ob ained
om e e y anspo expe imen . Op ical conduc i i y also showed a ais-
ing edge a ω∼500 meV [123], again oo la ge compa ed o anspo gaps.
The ac ha he gap de i ed om ARPES and op ical conduc i i y is much
la ge han he ones ob ained om anspo emains an unsol ed p oblem
o da e, p ecluding any consis en band-s uc u e unde s anding o hese
ma e ials.
Finally, much less is known abou (TaSe4)3I [16,120,124], which has gene -
ally been assumed o beha e like i s Nb coun e pa . The D4h →D2d s uc-
u al phase ansi ion was measu ed a a 200 K [124], bu he S4phase has
no been epo ed. Ve y ecen ly, he D2d phase was epo ed o be me allic
wi h a esis ance kink-pla eau a 150 K, a e omagne ic ansi ion a 8K,
and a supe conduc ing one a Tc=3K [36], in s a k con as o p e ious
obse a ions. The coexis ence o e omagne ism and supe conduc i i y is
a a ely epo ed phenomenon [37,38], and i is o en aken as a hin ha
pai ing could be in a uncon en ional odd-pa i y iple channel. While ex-
pe imen s on supe conduc i i y a e a a e y ea ly s age, a band s uc u e
3.2 me hods & heo e ical esul s o (MSe4)3I mcs.66
Figu e 3.2: (a) A omic s uc u e o (TaSe4)3I in he D4h phase and (b) B illouin
zones o he a omic s uc u e o in (a) shown in blue, and o he
app oxima e e ec i e s uc u e desc ibed in he ex which con ains
one o mula uni
unde s anding o he basic p ope ies o (TaSe4)3I is clea ly needed as a
s a ing poin o unde s and hese unusual beha io .
3.2 me hods & heo e ical esul s o (MSe4)3I mcs.
In he p esen Sec ion we will p esen ou heo e ical and compu a ional
amewo k and me hodology along he esul s ob ained by applying said
me hodology. These esul s will se e as heo e ical suppo o me ge he
pas esea ch p esen ed in Sec. 3.1.5and lay g ound o he pos e io expe -
imen al wo k p esen ed in Sec. 3.3.
3.2.1DFT de ails and me hodology
Wi h he aim o explaining he p e iously discussed phenomenology, i s -
p inciples DFT band s uc u e calcula ions o he di e en epo ed s uc-
u es we e pe o med using VASP [67,68] .6.2.1wi h p ojec o -augmen ed
wa e pseudopo en ials using wo app oxima ions: he gene alized g adi-
en app oxima ion wi h PBE pa ame iza ion [126] and he mBJ meh od
[78]. The PBE unc ional se es as a eliable and compu a ionally unexpen-
si e baseline o s uc u al and quali a i e elec onic p ope ies, while he
mBJ po en ial is known o yield imp o ed band gap es ima ions7by be -
7LDA and GGA signi ican ly unde es ima e band gaps since hey do no ully cap u e
he discon inui y in he exchange-co ela ion po en ial. Mo eo e , hey su e om sel -
in e ac ion e o s which a i icially delocalizes elec ons and educes he e ec i e bandgap.
By doing so, mBJ in oduces a semilocal co ec ion o he exchange e m and empi ical
pa ame iza ion. mBJ e ec i ely mimics he beha iou o he exac exchange po en ial o
3.2 me hods & heo e ical esul s o (MSe4)3I mcs.67
e app oxima ing he exac exchange po en ial. We employ bo h me hods
in o de o explain he possible me allic beha iou o (TaSe4)3I in he D2d
phase and ge accu a e bandgaps o he di e en phases. Fig. 3.2shows
he c ys al s uc u e o (MSe4)3I compounds in he D4h phase. The MSe4
uni s o m one dimensional chains, whe e each M a om is sandwiched be-
ween Se4 ec angula uni s and sepa a ed by I−ions. Each Se4adjacen
ec angula uni is o a ed a ound 45◦and modula ed by small dis o ions
on Se posi ions. The ypical uni cell o hese compounds is e agonal
(a=b=c) and con ains wo me allic chains. The pa icula case o he
Fig. 3.2is (TaSe4)3I in D4h phase, wi h a=b=9.719 Å and c=19.363 Å.
The di e ence be ween phases i.e D4h,D2d and S4, is gi en by he ela i e
dis ance be ween M a oms and small modula ions in he Se posi ions.
Fo (NbSe4)3I, a de ailed s uc u al cha ac e iza ion is a ailable o he
h ee conside ed phases D4h [17], D2d [101], and S4[104]. The e o e, all
DFT calcula ions o hese compounds we e pe o med using he expe i-
men ally measu ed la ice pa ame e s. Sel -consis en calcula ions consid-
e ing spin-o bi coupling we e ound o be well con e ged o a kine ic
ene gy cu o o 520 eV and a 9×9×5k-mesh sampling. Con e sely, in he
case o (TaSe4)3I, no de ailed cha ac e iza ion exis s. Gi en he simila i y
wi h he Nb compounds [101], we ob ained he s uc u es o he T com-
pounds pe o ming a s uc u al elaxa ion s a ing wi h he posi ions o he
expe imen ally measu ed Nb-based compounds [18,127]. The elaxa ion
calcula ions we e pe o med using a conjuga e g adien algo i hm [128]
and we e ound o be well con e ged o a kine ic ene gy cu o o 520 eV
and a 11 ×11 ×5k-mesh sampling. Fo phases D4h and D2d, he s a ing
poin we e he p e- elaxed s uc u es in Re . [127] while, o phase S4, a di-
ec elaxa ion wi h he conjuga e g adien algo i hm was pe o med om
he o iginal posi ions o he isoelec onic Nb compound in he same phase
since no p e- elaxed da a was a ailable. This s a egy p io i izes he con-
se a ion o he space-g oup symme ies om he o iginal Nb s uc u es.
Sel -consis en calcula ions conside ing spin-o bi coupling we e ound o
be well con e ged o a 520 eV kine ic ene gy cu o and a 11 ×11 ×5k-
mesh. Densi y o s a es calcula ions we e pe o med using a 15 ×15 ×9
k-mesh and a ene gy esolu ion o 0.7meV.
3.2.2DFT band s uc u es.
The elec onic band s uc u es o (NbSe4)3I and (TaSe4)3I a e p esen ed in
Fig. 3.3in bo h PBE and mBJ app oxima ions. The band s uc u es o bo h
Nb and Ta compounds a e o e all simila o all phases. A se o 8low
ene gy bands is obse ed in he ene gy window E∈[−0.5,0.5]eV. This se
semiconduc o s and insula o s and i is cheape compu a ionally han hyb id unc ionals
o GW app oxima ion.
3.2 me hods & heo e ical esul s o (MSe4)3I mcs.68
S uc u e a(Å)c(Å)1
6−dMi−Mi+1
c2
(×10−4)∆d Gap PBE (eV) Gap mBJ (eV)
Nb
D4h 9.4891 19.1323 0.11 -0.11 -0.46 -0.11 -0.11 -0.43 1.15 0.09 0.26
D2d 9.4500 19.0799 0.00 -0.47 -0.39 -0.00 -0.47 -0.39 1.32 0.18 0.31
S49.4365 19.0461 1.07 -0.17 -0.12 -0.58 -0.00 -1.01 1.71 0.28 0.40
Ta
D4h 9.7192 19.3626 0.09 -0.09 -0.37 -0.09 -0.09 -0.37 1.05 0.11 0.27
D2d 9.4365 19.4365 0.23 -0.00 -0.26 -0.23 -0.00 -0.28 1.00 0.06 0.19
S49.4365 19.0461 0.00 -0.39 -0.26 -0.01 -0.33 -0.30 1.13 0.12 0.24
Table 1: La ice pa ame e s, me al-me al dis ances, dis o ion ∆d and bandgaps o PBE and mBJ app oxima ions and hyb id po en ial o
he di e en s uc u es conside ed in he ex .
3.2 me hods & heo e ical esul s o (MSe4)3I mcs.69
is ound close o he alence bands in he Nb compounds compa ed o he
Ta compounds. These bands ha e a dominan o bi al weigh coming om
dz2o bi als o he Ta a oms, as shown in he densi y o s a es in Fig. 3.4
and as explained o iginally by G essie e . al. [103,120]. The ac ha hese
bands ha e dominan weigh in a single Ta o bi al sugges s a simple igh
binding model should desc ibe hese bands co ec ly, as shown in he ol-
lowing Sec ion.
Rega ding he band gaps, in he case o (NbSe4)3I, he mBJ me hod shows
a g ea e band gaps han he PBE pa ame iza ion o he h ee phases. In
PBE, he band gap is sligh ly indi ec , wi h he alence band maximum
sligh ly o Γin he M−Γdi ec ion and he conduc ion band minimum
a Γ. In mBJ, he band gap is also indi ec o D2d and S4phases, bu i is
di ec o D4h. Simila ly, o (TaSe4)3I he mBJ me hod also shows a g ea e
band gap han PBE o all phases. Bo h in mBJ and PBE app oxima ions
he band gaps a e di ec be ween Γ−Γ o in e sion conse ing phase D4h
and sligh ly indi ec o in e sion-b oken phases D2d and S4.
In o de o unde s and he di e en alues o he ob ained om DFT
calcula ions we ecall ha he ela i e dis ances be ween Nb a oms we e
epo ed o be ela ed o he elec onic band gap in he pas . In he in e es
o gaining insigh o his s a emen o (MSe4)3I compounds we in oduce
he concep o ime iza ion dis o ion8∆d,
∆d =
u
u
6
X
i=11
6−dMi−Mi+1
c2
(3.5)
whe e dMi−Mi+1=ˆ
diis he dis ance be ween neighbou ing M a oms along
he cdi ec ion, c=|c|and he sum uns o e he 6M a oms in he
(MSe4)3I compounds. The quan i y ∆d measu es how a a 6-a om chain
is om being e enly spaced, and anishes when all dis ances be ween M
a oms a e equal, i.e. ˆ
di=ˆ
dj o all i,j. A summa y o ele an s uc u al
da a and dis o ions, along wi h he band gaps ob ained bo h in he PBE
pa ame iza ion and mBJ me hod a e p esen ed in Table 1. The esul s in
he able sugges an ela ion be ween dis o ion, cell olume and elec onic
band gap. In gene al e ms, a g ea e dis o ion leads o wide elec onic
band gaps. E en so, he cell olume also seems o plays a ole in enhanc-
ing o educing he band gap, since smalle cell olumes lead o smalle
8T ime iza ion e e s o a s uc u al dis o ion in he quasi-1D chains o he TMCs, whe e
he spacing be ween adjacen me al a oms along he chain becomes unequal. This dis o -
ion c ea es a pe iodic al e na ion in he dis ances be ween neighno ing me al a oms, e ec-
i ely g ouping hem in o se s o h ee wi hin he uni cell, hence he name. T ime iza ion
a ec s he elec onic s uc u e by opening a band gap in he ma e ial. I is a key ea u e
dis inguishing he semiconduc ing beha io o hese compounds and plays a c i ical ole
in hei anspo and op ical p ope ies.
3.2 me hods & heo e ical esul s o (MSe4)3I mcs.70
band gaps. The da a ecollec ed in Sec. 3.1.5 o anspo band gaps in
Nb-compounds, shows he ollowing end: ∆ED4h > ∆ES4 > ∆ED2d, while
ou da a sugges s ha ∆ES4 > ∆ED2d > ∆ED4h. E en hough he ends a e
dissimila , he magni udes o he DFT elec onic gaps a e compa able o
he ones ob ained wi h anspo measu emen s.
3.2.3Tigh -Binding models
Since he o bi al- esol ed densi y o s a es in Fig. 3.4shows ha he bands
nea he Fe mi le el a e domina ed by M dz2o bi als, we can gain a deepe
unde s anding o he s uc u al dependence o he band gap om a sim-
ple igh -binding model con aining only such o bi als in Ta si es. The model
will be cons ained by he symme ies o each phase: D4h is gene a ed by in-
e sion I, he glide {m110|1
2
1
2
1
2}, he o oin e sion S4and he wo- old sc ew
{2100|1
2
1
2
1
2}. B eaking o in e sion hen leads o he poin g oup D2d and u -
he b eaking he glide and wo- old axis leads inally o poin g oup S4.
In he phase o highe symme y, D4h, he e a e only wo non-equi alen M
a oms, as shown in Fig. 3.5a) and b). The simples model he e o e only con-
ains wo di e en on-si e po en ials ∆1and ∆2, wo in achain hoppings 1
and 2and a single in e chain hopping ⊥. As he igu e shows, his model
ac ually has an acciden al ansla ion symme y, wi h a educed uni cell
wi h 3M si es, ep esen ing a single o mula uni (MSe4)3I ( he ue uni
cell would ha e 4 o mula uni s). The la ice pa ame e s o his educed
uni cell a e c∗=c
2and a∗=a
√2, and he co esponding enla ged B illouin
Zone BZ∗is shown in Fig. 3.2, wi h high symme y poin s deno ed as A∗,
Z∗and so on. The Hamil onian o his model is
H= −
∆1+ ⊥ (k∥) 1eikzc/6 3e−ikzc/6
1e−ikzc/6 ∆2+ ⊥ (k∥) 2eikzc/6
3eikzc/6 2e−ikzc/6 ∆3+ ⊥ (k∥)
(3.6)
whe e 1= 2and
(k∥) = 2cos(kx+ky)a
2√2+cos(kx−ky)a
2√2.
To accoun o he ime iza ion dis o ion, we se 2=T−δ/2 and 3=T+δ
so ha he a e age hopping is and δpa ame izes he ime iza ion dis o -
ion assuming ha he hoppings change linea ly wi h he dis ance be ween
o bi als.
The p esence o he ex a ansla ion symme ies in his model is acciden al,
as he inclusion o u he neighbo in a- and in e chain hoppings would
3.2 me hods & heo e ical esul s o (MSe4)3I mcs.71
Figu e 3.3: Ab-ini io bands uc u es o he (MSe4)3I compounds in he h ee di e en s uc u es D4h,D2d and S4 om le o igh . Top
ow (a-c) shows (NbSe4)3I, bo om ow (d- ) shows (TaSe4)3I. Solid back lines we e ob ained using PBE pseudopo en ials and
dashed blue lines we e ob ained using mBJ hyb id po en ials.
3.2 me hods & heo e ical esul s o (MSe4)3I mcs.72
−0.6−0.4−0.2 0.0 0.2 0.4 0.6
Enegy [eV ]
0
2
4
6
8
10
12
14
DOS a.u.
To al DOS
Ta d2
z
Se px
Se py
Figu e 3.4: O bi al esol ed densi y o s a es o (TaSe4)3I in he D2d phase com-
pu ed wi hin he PBE pa ame iza ion. This densi y o s a es co e-
sponds o he bands shown in Fig. 3.3(e). The bands nea he Fe mi
le el ha e dominan dz2o bi al cha ac e on he Ta si es. This is ue
o all compu ed band s uc u es (no shown).
indeed equi e he use o he ull 12 si e uni cell. This u he neighbo
hoppings a e howe e expec ed o be smalle , so ha he model in Eq. 3.6
se es as a good i s app oxima ion o he band s uc u e. Physically, his
means ha he ac ual posi ions o he Se4uni s and I−ions has li le e ec
on he low ene gy M-de i ed bands.
Fig. 3.6(a) shows he bands ob ained om Eq. 3.6 o ∆i=0, =1eV,
δ=0.3eV and ⊥=0.05 eV. Fo compa ison, we also show he same plo
o δ=0, and we see ha a ini e dis o ion δopens a gap a he Fe mi
le el. A ini e alue o ∆iwould u he con ibu e o he opening o he
gap and is no conside ed o simplici y. We also obse e ha he p esence
o small in e chain hopping leads o an indi ec gap, wi h he alence band
maximum a A∗and he conduc ion band minimum a Z∗.
To compa e he igh -binding model wi h he compu ed ab-ini io band
s uc u es, in Fig. 3.6(b) we also plo he bands in he physical 12 si e
uni cell. Doing so we obse e a back olding o he bands, so ha below
he Fe mi le el we now ha e ou bands a Γ, o igina ing om A∗,R∗,M∗
and Γ. The gene al ag eemen o he olded bands wi h he ab-ini io band
s uc u es in Fig. 3.3sugges s ha his educed model is indeed a e y
3.2 me hods & heo e ical esul s o (MSe4)3I mcs.79
Figu e 3.9: Ab-ini io low ene gy band s uc u e (le ) and o al DOS ( igh ) ob-
ained using PBE app oxima ion o (a) Ta D4h phase and (b) Ta D2d
phase. No e a Van Ho e singula i y o holes ma ked wi h a do ed line
in bo h cases.
unin en ional doping o he samples. This could ha e occu ed because he
samples did show some non-s ochiome y [36]. I his is he case, one migh
ask whe he he e is any di e ence be ween doping elec ons o holes, in
pa icula ega ding he e omagne ic and supe conduc ing ins abili ies.
To answe hese ques ions, we ha e compu ed he DOS o he (TaSe4)3I
D4h and D2d s uc u es, shown in Fig. 3.9. This e eals ha while doping
wi h elec ons leads o a a he smoo h inc ease o he ca ie s, doping wi h
holes leads o a as e aise up o a Van Ho e singula i y a ED2d
VH ≈−0.07
eV and ED4h
VH ≈−0.09 eV o he wo phases. This singula i y eme ges due
o a change o shape o he Fe mi su ace om con ex o conca e as he
ene gy is lowe ed, as shown in Fig. 3.10.
3.2 me hods & heo e ical esul s o (MSe4)3I mcs.80
Figu e 3.10: (a) Fe mi su ace o phase D4h a E= −0.16 eV. The plane kz=0is
shown in blue. (b) kz=0Sec ion o he Fe mi su ace o phase D4h
a di e en ene gies wi h espec o he Van Ho e singula i y: sligh ly
below (dashed), a he singula i y (do ed-dashed) and sligh ly abo e
(s aigh ).
The p esence o his Van Ho e singula i y in he alence band sugges s ha
a e omagne ic ins abili y could be igge ed by a small amoun o hole
doping, as i is p edic ed o occu in GaSe [136] and simila monochalco-
genides [137–139]. Van Ho e singula i ies in gene al display di e en ypes
o ins abili ies due o he enhanced densi y o s a es [140], and hei gene ic
phase diag ams include bo h e omagne ic s a es and uncon en ional pai -
ing [141,142] media ed by he epulsi e Coulomb in e ac ions [143]. While
mo e wo k is needed o measu e he amoun and ype o doping, as well
as he Fe mi su ace shape, we belie e he exis ence o his Van Ho e sin-
gula i y is a unique ea u e o he hole doped sys em which we conjec u e
will play a ole in he low empe a u e ins abili ies.
3.2.7Concluding ema ks on he heo e ical esul s.
By p o iding he i s de ailed band s uc u e cha ac e iza ion o he (MSe4)3I
compounds, ou analysis has e ealed ha despi e ha ing a e y complex
la ice s uc u e wi h 64 a oms in he uni cell, hei low ene gy elec onic
s uc u e is ac ually e y simple. I can be unde s ood in e ms o a quasi
1D e ec i e model o dz2o bi als in a chain wi h a h ee si e uni cell, and
a ime iza ion ha gi es ise o a gap [103,120]. Ou de ailed cha ac e iza-
ion has explained a numbe o puzzles in he li e a u e, and will se e o
p ope ly in e p e u u e expe imen s in his class o ma e ials.
Fi s , we ha e p o ided a quan i a i e p edic ion o he anspo gaps
o he di e en s uc u es, which a e b oadly consis en wi h expe imen al
3.3 expe imen al ealiza ion: (nbse4)3i.81
obse a ions. Ou esul s show ha he gap magni ude is no only co e-
la ed wi h he amoun o ime iza ion in he di e en s uc u es, bu also
wi h he uni cell olume. In he u u e, ou p edic ions will also be ele-
an o con i m he p oposed dis inc ion be ween ype I and II samples in
anspo expe imen s, and o cla i y he anspo p ope ies o he S4low
empe a u e phase.
Second, ou wo k p o ides a clea esolu ion o he disc epancy be ween
he gaps epo ed in anspo s. hose in ARPES and op ical conduc i -
i y. These expe imen s we e ca ied ou in he high empe a u e D4h phase,
whose band s uc u e is essen ially ha o an indi ec gap semiconduc o
wi h alence band a A∗and conduc ion bands a Z∗. A e y weak modu-
la ion olds bo h bands o Γ, bu obse ing hem in ARPES o p obing an
op ical ansi ion be ween hem is ex emely ha d due o he e y low spec-
al weigh , p opo ional o he weak modula ion. The la ge gaps quo ed
in ARPES and op ical conduc i i y ac ually co espond o he di ec gap
in he un olded bands, which is much la ge han he ue gap. Ou p e-
dic ions can be eadily es ed in new ARPES expe imen s p obing he A∗
poin . In addi ion, u u e low empe a u e measu emen s in he D2d phase
migh be mo e sensi i e o de ec he ue gap in op ical conduc i i y and
ARPES, as we ha e shown.
Finally, ou wo k calls o mo e s udies o unde s and he o igin o he
me allic beha iou o D2d (TaSe4)3I s udied in Re . [36]. Bo h ARPES and
op ical conduc i i y will be use ul o quan i y he exis ence o any ex in-
sic doping. In addi ion, i he doping is hole-like, ARPES expe imen s can
di ec ly map he Fe mi su ace and con i m he exis ence o a low ene gy
Van Ho e singula i y, which will be ele an o unde s and he magne ic
and supe conduc ing ins abili ies. We hope ou wo k will mo i a e u he
s udies on he subjec o explain his unusual coexis ence.
3.3 expe imen al ealiza ion: (nbse4)3i.
The heo e ical insigh s p o ided in he p e ious Sec ion ha e laid he
ounda ion o he expe imen al collabo a ion p esen ed in his Chap e .
Building upon hese p edic ions, he expe imen al wo k ca ied ou by he
g oup o N. Sch ö e ocuses on e i ying he heo e ical claims and un-
co e ing addi ional phenomena linked o he in e play o band s uc u e
and op oelec onic p ope ies. By employing ad anced ARPES echniques,
hei expe imen al wo k del es in o he spec al weigh dis ibu ions ac oss
he h ee-dimensional B illouin zone, o e ing a comp ehensi e iew o he
elec onic s a es. This app oach no only co obo a es he heo e ical ind-
ings bu also ex ends ou unde s anding o how app oxima e da k s a es
in luence he op ical and anspo beha io s o (NbSe4)3I and i s amily.
3.3 expe imen al ealiza ion: (nbse4)3i.82
The impo ance o his Sec ion lies in i s con ibu ion o b idging he gap
be ween heo e ical p edic ions and expe imen al obse a ions. The expe -
imen al esul s p esen ed he ein p o ide c ucial e idence o he impac
o app oxima e ansla ional symme ies and spec al weigh modula ions
on he op oelec onic p ope ies o (NbSe4)3I and i s amily. These esul s
deepen ou unde s anding o he ma e ial’s band s uc u e and pa e he
way o u u e s udies on ela ed compounds. Ul ima ely, his wo k high-
ligh s he syne gy be ween heo y and expe imen in add essing complex
p oblems in condensed ma e physics and unde sco es he signi icance
o (NbSe4)3I as a model sys em o explo ing undamen al concep s in low-
dimensional ma e ials. In he ollowing pages we will ocus mo e on unde -
s anding he expe imen al esul s and hei in e play wi h ou heo e ical
indings a he in he expe imen al de ails, o hose we e e o he o iginal
a icle, Re . [144].
As a no el insigh , his expe imen al wo k iden i ies he modula ions dis-
cussed in he p e ious Sec ion as mani es a ions o app oxima ely da k s a es,
d awing on he amewo k in oduced in he ecen wo k by he g oup o
K. Su Kim [145]. In he ollowing Sec ion, we in oduce he concep o da k
s a es and p esen a de ailed analysis o he expe imen al esul s based on
his in e p e a ion, he eby ex ending ou heo e ical insigh s and b oaden-
ing he scope o he de eloped amewo k.
3.3.1In oduc ion: Wha a e da k s a es.
Quan um s a es ha do no allow o op ical ansi ions a e o en cha -
ac e ized as da k [146–148]. Al hough no uni e sal amewo k exis s o
hei o igin, a ecen publica ion has p oposed ha such s a es commonly
a ise in c ys als whe e wo subla ices a e ela ed by a glide symme y.
This symme y leads o double des uc i e in e e ence in he op ical ma-
ix elemen s ac oss al e na ing B illouin zones, e ec i ely supp essing
pho oemission signals and p e en ing hese s a es om being obse ed
in ARPES [145]. No ably, his classi ica ion appea ed only a e he wo k
p esen ed in he p e ious Sec ions had been comple ed. None heless, he
low-spec al-weigh elec onic s a es we iden i ied (o igina ing om band
olding and o bi al cha ac e associa ed wi h he quasi-1D chain geome-
y), exhibi ema kably simila phenomenology. In e ospec , hese ea-
u es can be na u ally in e p e ed wi hin he amewo k o app oxima ely
da k s a es, p o iding a aluable concep ual ool o analyze he expe imen-
al esul s p esen ed below.
While Re . [145] emphasizes he impo ance o glide symme ies o he ap-
pea ance o da k s a es, as discussed in he p e ious Sec ion, app oxima e
3.3 expe imen al ealiza ion: (nbse4)3i.83
ansla ion symme ies a e o en esponsible o he supp ession o spec-
al weigh and, consequen ly, op ical ansi ions appea ing in ARPES. As
explained in Sec. 3.2.3, app oxima e ansla ional symme ies a ise when
he lowes -o de e ms in an e ec i e igh -binding model allow o ans-
la ional symme ies in ol ing a small uni cell. These symme ies a e hen
b oken by highe -o de e ms ha equi e ex ending he uni cell o i s ull
p imi i e size [149]. As shown in Sec. 3.2.4, hese highe -o de co ec ions
lead o he eme gence o olded bands wi h small spec al weigh alongside
b igh bands wi h s ong spec al weigh ha ollow he symme y o he
smalle uni cell. In he limi ing case, he comple e supp ession o olded
bands would only be possible when he app oxima e ansla ion symme-
ies become exac symme ies o he c ys al s uc u e. In such a case, he
sys em can be desc ibed by a smalle uni cell and ewe bands.
In he ollowing pages, he expe imen al wo k ca ied ou by ou collab-
o a o s om N. Sch ö e g oup con i ms expe imen ally wha we al eady
demons a ed heo e ically: ha wha we now call app oxima e da k s a es,
which esul om app oxima e ansla ion symme ies, can in luence he
op oelec onic p ope ies by edis ibu ing he spec al weigh and causing
a di ec band gap semiconduc o (NbSe4)3I o beha e as i i had an indi ec
band gap. In he p e ious Sec ion, we concluded ha op ical ansi ions in
his ma e ial a e domina ed by b igh s a es wi h s ong spec al weigh
obeying he symme y o a smalle app oxima e uni cell, esul ing in an
enhanced op ical band gap compa ed o he anspo gap, in he ollowing,
we will es his conclusion by analyzing ARPES esul s.
While p e ious ARPES expe imen s ailed o cap u e his beha io p op-
e ly, due o ou heo e ical indings, ou expe imen al colleagues obse e
hese app oxima e da k s a es by inspec ing he A∗poin in ecip ocal space.
By using pho on-ene gy-dependen ARPES measu emen s, ou collabo a-
o s ound ha he elec onic s uc u e o (NbSe4)3I de ia es om he p e-
iously assumed quasi-1D na u e and ins ead displays s ongly dispe si e
bands along all h ee momen um di ec ions. By ob aining he elec onic
band s uc u e along all high-symme y kpa hs in a 3D B illouin zone
co esponding o he smalle app oxima e uni cell con aining only 3Nb
a oms, we can iden i y a spec al weigh modula ion esul ing in app oxi-
ma ely da k s a es. Speci ically, he da a collec ed by ou collabo a o s sup-
po s a ela i ely small di ec anspo gap o 0.2eV in (NbSe4)3I ha
allows o he he mally ac i a ed occupa ion o app oxima ely da k s a es,
which can con ibu e o anspo esponses. Howe e , hey show ha he
op ical ansi ions a e domina ed by he b igh s a es de ec ed in ARPES,
which o m an indi ec band gap.
Beyond he speci ic example o (NbSe4)3I, hese expe imen s unde line
3.3 expe imen al ealiza ion: (nbse4)3i.84
he impac o app oxima e symme ies on spec oscopic measu emen s,
demons a ing how he p esence o app oxima e da k s a es leads o a
di ec - o-indi ec band gap ansi ion ha de e mines he op oelec onic
p ope ies o semiconduc o s.
3.3.2De ailed s uc u al classi ica ion.
The main s uc u al de ails o (MSe4)3I(M=Ta,Nb) TMCs a e p esen in
Sec. 3.1.4and Fig. 3.2. The Nb alence s a es a e sequen ially 4+,4+,5+,
wi h a sho e bond leng h o he Nb4+−Nb4+bond (3.02Å s. 3.25Å o
he Nb4+−Nb5+bond ). As shown in he p e ious Sec ion, he e is an
app oxima e symme y shown in Fig.3.11 (a) which is a uni cell wi h 1/4
he olume and jus 3Nb a oms. The uni cells a e ela ed by c∗=c/2 and
a∗=a/√2.
In line wi h p e ious bibliog aphy [123], ou collabo a o s ound he clea -
age plane10 o be he (110)plane. In his geome y, as shown in Fig. 3.11 (b),
we de ine kch ≡kzas a k-pa h along he chain, kip ≡kxas he di ec ion
accessed by in-plane o a ion o he analyse wi h espec o he sample
no mal, and k⊥≡kyas he di ec ion no mal o he su ace. In k-space, he
c ys allog aphic 12-Nb BZ is he one ma ked wih a dashed blue line in
Fig. 3.11 (b), bu he BZ co esponding o he app oxima e 3-Nb uni cell is
la ge and o a ed as shown in Figs. 3.2and 3.11, he e ma ked in ed. Each
o he Γ∗,Z∗,M∗,A∗poin s map o Γpoin s o he c ys allog aphic uni cell.
3.3.3App oxima ely da k s a es as seen in ARPES.
The ARPES dispe sion along he chain di ec ion, kch is p esen ed in Fig.
3.12 (a). The main no able ea u e o said dispe sion is ha he spec al
weigh is domina ed by a b igh band whose maximum is loca ed a ≈0.5eV
below EFand p esen s a pe iodici y o 4π/c. Hence, he b igh es ea u es
in he da a seem o ollow he expec ed pe iodici y o he app oxima e
3-Nb uni cell, no he c ys allog aphic 12-Nb one. Mo eo e , his da a is
cong uen wi h olde li e a u e epo s, albei ha he alence band maxi-
mum o he main band was ound sligh ly a highe binding ene gies [123,
125].
Along he b igh band, weake bands abo e EF−0.5eV a e also obse ed
in Fig. 3.12 (a), ma ked by a black a ow. E en hough hese a e ue eigen-
s a es o he ull sys em, hey a e app oxima ely da k since hei spec al
weigh is supp esed due o app oxima e ansla ion symme ies. This sig-
10 The clea age plane is he speci ic c ys allog aphic plane along which a c ys al sample is
clea ed o c ea e he su ace o measu emen .
3.3 expe imen al ealiza ion: (nbse4)3i.85
Nb
I
Se
(a) (b)
3.25Å 3.25Å 3.06Å
4+ 5+ 4+ 4+ 5+ 4+
b
a
c
(110)
(001)
3.25Å 3.06Å
(c)
X*
Z*
A*
Г*
M*
R*
c
d
(d)
R*
X
e
Nb
I
Se
(a) (b)
3.25Å 3.25Å 3.06Å
4+ 5+ 4+ 4+ 5+ 4+
b
a
c
(110)
(001)
3.25Å 3.06Å
(c)
X*
Z*
A*
Г*
M*
R*
c
d
(d)
R*
X
e
Nb
I
Se
(a) (b)
3.25Å 3.25Å 3.06Å
4+ 5+ 4+ 4+ 5+ 4+
b
a
c
(110)
(001)
3.25Å 3.06Å
(c)
X*
Z*
A*
Г*
M*
R*
c
d
(d)
R*
X
e
kz≡kch
kx≡kip
ky≡k⊥
Figu e 3.11: (a) Simpli ied 12-Nb uni cell (blue line) and app oxima e 3-Nb uni
cell ( ed line), see Fig. 3.5. (b) App oxima e 3-Nb uni cell’s BZ wi h
high simme y poin s ma ked wi h * o dis inguish om ini ial 12-Nb
uni cell’s BZ. g ay shaded a eas c,d, and e ep esen he Fe mi su ace
maps shown la e . Adap ed om Re . [144].
nals ha he 12-Nb uni cell mus be conside ed in o de o iden i y all
s a es wi hin he measu ed spec al unc ion, and so ca e ul unde s and-
ing o he in e play o he c ys allog aphic and app oxima e uni cells is
equi ed. Fig. 3.12(b) shows he DFT11 bands uc u e o D4h phase. The
low ene gy bands, mainly cha ac e ized by Nb’s 4d2
zo bi als ha e a cosine-
like dispe sion along kch wi h a pe iodici y o 4π/c o 2π/c∗, ollowing he
pe iodici y o he app oxima e 3-Nb uni cell and no he o e all 2π/c pe-
iodici y om he c ys allog aphic la ice. The e is a mul iplici y in hese
cosine-like dispe sions de i ed om he combina ion o ini e hoppings
ans e se o he chain di ec ion oge he wi h he expansion o he uni
cell when compa ed wi h he 3-Nb desc ip ion. Howe e , only one o hese
b anches is b igh expe imen ally, while he h ee o he a e app oxima ely
da k s a es. This aligns wi h he 3-Nb pic u e, whe e only a single cosine-
like band would be p essen dispe sing upwa ds along kch om a mini-
mum a Γ∗. The es o hem appea due o pa ial b eaking o 3-Nb uni
cell ansla ional symme y.
Since ARPES spec al weigh seems o ollow he 3-Nb app oxima e cell de-
sc ip ion (see App. C.4 o he de ails on how Fig. 3.13 (b) was cons uc ed),
in Fig. 3.13 we can inally compa e ou collabo a o s’ ARPES da a wi h an
un olded DFT calcula ion ob ained by ollowing he me hods p esen ed in
App. C.1 ha we al eady used in Sec. 3.2.4. The dis ibu ion o spec al
11 All DFT calcula ions o his expe imen al Sec ion we e ca ied ou unde he same guide-
lines as hose p esen ed in Sec. 3.2.1,i.e. wi h p ojec o -augmen ed wa e pseudopo en ials,
GGA app oxima ion wi h PBE pa ame iza ion [74] using a 520eV kine ic ene gy cu o
and a 9×9×5 k−mesh.
3.3 expe imen al ealiza ion: (nbse4)3i.86
(a) (b) (c)
app oxima ely da k s a e
Figu e 3.12: App oxima ely da k s a es e ealed in dispe sion o spec al unc ion
along chain. (a) Dispe sion along he chain di ec ion. The ene gy cu
is measu ed a 43eV and 90K, ed a ows ep esen he pe iodici y o
4π/c and he black a ow indica es he app oxima ely da k s a es. (b)
DFT calcula ion o 12-Nb a om s uc u e uni cell. Blue band co e-
spond o he b igh bands shown in (a). Adap ed om Re . [144].
weigh ag ees well wi h he un olded DFT bands uc u e and pa icula ly
highligh s ha b igh band maximum in he 3D BZ is ound a he A∗poin ,
which can only be accessed expe imen ally by he co ec combina ion o
kpe p and kch (see App. C.4), a sub le y which was no app ecia ed in p e-
ious ARPES epo s. This would explain he incohe en pic u e p esen ed
in Sec. 3.1.4and add essed in Sec. 3.2.4.
ARPES measu emen s we e pe o med a 90 K, which should ende he
sys em’s s uc u e o he D2d phase [16]. Due o he issue o he mally-
ac i a ed loss o iodine om he su ace, as p e iously s udied in (TaSe4)2I
in Re . [150] whe e sample cha ging is an issue a low empe a u es, a ull
s udy on he empe a u e dependence o band gap was no possible. As
a esul , o he p obes should be be e sui ed o s udying he empe a-
u e e olu ion o he gap [119]. Ne e heless, ou collabo a o s’ expe imen
p obed bulk-like s a es in he measu emen and hence he main conclu-
sions a e alid i espec i e o he space g oup, since he D2h phase has also
a12-Nb uni cell.
3.3 expe imen al ealiza ion: (nbse4)3i.87
(a) (b)
Г* Z* R* A* M* Г* Z* R* A* M*
s ong op ical ansi ion
app oxima ely da k s a e
Figu e 3.13: Compa ison be ween un olded bands uc u e and ARPES e ealing
b igh and da k s a es. (a) Un olded DFT calcula ion ( ollowing he
me hod p esen ed in App. C.1& Sec. 3.2.4). The physical c ys al s uc-
u e wi h 12 Nb a oms a e in e p e ed as a pe iodically epea ed 3Nb
a oms one. (b) ARPES dispe sion along he equi alen kpa h. Black
a ows indica e app oxima ely da k s a es. Adap ed om Re . [144].
3.3.4Discussion & conclusion on he expe imen al esul s.
Ou collabo a o s’ expe imen al da a plo ed in Fig. 3.13 (b) shows ha
he band s uc u e o (NbSe4)3Iis e y close o a di ec band gap a he
A∗and Z∗poin s. Howe e , due o he spec al weigh modula ion o he
app oxima ely da k s a es clea ly de ec ed wi h ARPES, he op ical p ope -
ies a e a he domina ed by ansi ions be ween bands a highe ene gies
wi ha a la ge spec al weigh , leading (NbSe4)3I o beha e as a e ec i e
indi ec band gap semiconduc o in op ics. This in e es ing beha iou o e-
shadowed in Sec. 3.2.5, migh be no ha uncommon since i can happen
as long as bo h conduc ion and alence bands de i e om he same o bi al
and Wycko posi ion in he uni cell, leading o he app oxima e ansla-
ion symme y o apply o bo h equally. Ano he ea u e wo h no icing is
he ollowing: e en i olding pe u ba ion does no align conduc ion and
alence band edges, i will s ill be he case ha he op ical and anspo
esponses will gene ally be di e en , al hough he ue gap will no be di-
ec in ha case. Mo eo e , i conduc ion and alence bands de i e om
di e en o bi als and/o Wycko posi ions, hen i migh be ha only one
o hem shows a da k s a e modula ion, o ha such modula ion is di e -
en o each, leading o he op ical ma ix elemen s o be ini e.
3.4 disco e y o a new quasi-1d mc:(Nb4Se15 I2)I2.88
In conclusion, he expe imen al wo k ca ied by he g oup o N. Sch ö e
e ec i ely p o es ou hypo hesis p esen ed in Sec. 3.2and u he de elops
he p oposed amewo k, sol ing he puzzling li e a u e behind (MSe4)3I
(M=Nb,Ta) TMCs in e ms o da k s a es. I does so by showcasing a simple
bu d ama ic example o how app oxima ely da k s a es may impac he op-
oelec onic p ope ies o a sys em and lead o gla ing disc epancies wi h
o he expe imen al p obes such as anspo . These disc epancies can be
eadily ad essed by mapping ou he da k bands spec al weigh in an
ARPES expe imen , which s ands now as he main p obe o unde s and-
ing such sys ems and ad ess simila p obelms in o he ma e ials beyond
(NbSe4)3I. In his las Chap e we conduc ed a compehensi e bibliog aphic
e iew, e ec i ely p oposed a heo e ical amewo k and me hods o ad-
d ess disc epancies in exis ing li e a u e, p oposing an expe imen al eal-
iza ion ha would back ou hypo hesis ha was inally p oben expe imen-
ally.
3.4 disco e y o a new quasi-1d mc:(Nb4Se15 I2)I2.
The p eceding Sec ions o his Chap e ha e explo ed he in ica e in e play
be ween heo y and expe imen in he s udy o quasi-1D TMCs. We ha e
seen how disc epancies in he li e a u e conce ning cha ge o de ing, elec-
onic s uc u e, and op ical esponses can be econciled h ough a com-
bina ion o DFT and a ge ed expe imen al in es iga ions. This me hod-
ological app oach allowed us o es ablish a heo e ical amewo k ha was
subsequen ly alida ed h ough expe imen al wo k om ou collabo a o s.
Building upon his pe spec i e, we now shi ou ocus o he disco e y
and cha ac e iza ion o a no el quasi-1D TMC, (Nb4Se15I2)I2.
While he ma e ial was ini ially unco e ed ia expe imen al echniques by
he g oup o D. P. Shoemake , i s elec onic and s uc u al p ope ies chal-
lenge con en ional expec a ions o quasi-1D sys ems, making i a e ile
g ound o u he heo e ical explo a ion. Unlike many o i s coun e pa s,
(Nb4Se15I2)I2does no exhibi CDW o de , despi e i s quasi-1D s uc u e
and elemen al composi ion. Ins ead, i p esen s a mode a e bandgap o
Eg≈0.6eV and a unique chi al s acking, ea u es ha dis inguish i om
o he niobium-based TMCs. This Sec ion aims o place he disco e y o
(Nb4Se15I2)I2wi hin he b oade con ex o TMC di e si y, examining how
i s s uc u al and elec onic p ope ies i wi hin he landscape o p e i-
ously s udied ma e ials. By le e aging DFT calcula ions, we p o ide insigh
in o i s band s uc u e, o bi al cha ac e , and anspo p ope ies, ein o c-
ing he c i ical ole o i s -p inciples me hods in mode n ma e ials science.
This s udy no only expands he axonomy o quasi-1D TMCs bu also
opens new ques ions ega ding he ela ionship be ween dimensionali y,
3.5 concluding ema ks and ou look 95
Fi s ly, in Sec. 3.2, we ha e syn hesized and uni ied con lic ing pas e-
sea ch on he (MSe4)3ITMC se ies wi h M=Nb,Ta. To achie e his, we
es ablished a heo e ical amewo k o unde s anding he elec onic p op-
e ies o quasi-1D TMC compounds, pa icula ly he (MSe4)nIse ies wi h
M=Nb,Ta. Th ough band s uc u e calcula ions wi h di e en app oxi-
ma ions we ha e con ex ualized and ca aloged he di e en bandgaps o
all he amily o compounds, iden i ying ime iza ion and cell olume as
he main ac o s a ec ing he bandgap magni ude. By la e inco po a ing
band un olding echniques and symme y conside a ions, we ha e been
able o di ec ly compa e ou esul s wi h p e ious expe imen al ARPES
and anspo da a, esol ing inconsis encies in epo ed band gaps and
Fe mi su ace ea u es by de e mining ha small s uc u al modula ions
lead o small spec al weigh modula ions and e ec i e indi ec band gap
semiconduc o op ical beha iou . These esul s p o ide a clea e heo e ical
unde s anding o pas anspo and op ical expe imen s and p edic s ha
p obing he A∗poin wi h ARPES would uni y esul s on hese compounds.
This s udy also calls o new s udies o unde s and he me allic beha iou
o D2d (TaSe4)3I.
Secondly, in Sec. 3.3, we ha e success ully applied he heo e ical ame-
wo k p esen ed in Sec. 3.2 o unde s and he expe imen ally obse ed in-
di ec bandgap op ical beha io o (NbSe4)3I. A compound ha DFT p e-
dic s o be a di ec bandgap semiconduc o . Thanks o he measu emen s
o he g oup om N. Sch ö e , we de e mine ha said unexpec ed be-
ha iou comes om spec al weigh modula ions caused by small s uc-
u al changes. To de e mine his, ou collabo a o s p ob ed he A∗and
Z∗poin s wi h ARPES, success ully iden i ying wha we de ine as app ox-
ima ely da k s a es. The amewo k p esen ed in Sec. 3.2is hus u he
expanded by ecognizing ha his beha io is no uncommon when con-
duc ion and alence bands o igina e om he same o bi al and Wycko
posi ion, leading o app oxima e ansla ion symme y ha applies o bo h
bands equally. These expe imen al indings open new a enues o s udying
and unde s anding quasi-1D TMCs, de ining ARPES as one o he p ima y
p obes o in es iga ing he obse ed beha io .
Thi dly, in Sec. 3.4we ha e b oadened he explo a ion o quasi-1D TMCs
and opened new a enues o esea ch. We ha e cha ac e ized and s ud-
ied he disco e y o a new quasi-1D TMC, (Nb4Se15I2)I2, disco e ed and
syn hesized ia CVT by ou colleagues and p edic ed by DFT o be a
semiconduc o wi h a mode a e band gap o app oxima ely 0.6eV. The
anspo measu emen s ca ied ou by he g oup o D. P. Shoemake con-
i med i s semiconduc ing beha io , wi h an ac i a ion ene gy o 0.1eV,
likely due o shallow de ec s a es. Unlike o he ma e ials in he (MSe4)nI
3.5 concluding ema ks and ou look 96
amily, (Nb4Se15I2)I2does no exhibi CDW o de , as con i med by low-
empe a u e X- ay di ac ion and di e en ial scanning calo ime y. This
dis inc ion aises new ques ions abou he ole o symme y and elec onic
in e ac ions in s abilizing CDWs in closely ela ed sys ems. Mo eo e , ou
wo k sugges s new di ec ions o u u e esea ch. The disco e y o (Nb4Se15I2)I2
opens he possibili y o uning i s band s uc u e ia chemical doping, ex-
e nal ields, o s ain o explo e eme gen phases beyond semiconduc ing
beha io . Fu he mo e, gi en he complexi ies o magne o anspo in e-
la ed ma e ials such as (TaSe4)2I, i emains an open ques ion whe he
(Nb4Se15I2)I2could hos non i ial opological s a es o uncon en ional
anspo p ope ies unde speci ic condi ions. Expanding he chemical
phase space o quasi-1D TMCs by a ying chalcogen and halogen subs i u-
ions may lead o he disco e y o new ma e ials wi h exo ic elec onic phe-
nomena. We hope his wo k will inspi e u he expe imen al and heo e -
ical s udies aimed a unco e ing no el quan um s a es in low-dimensional
sys ems.As an in iguing example o he po en ial applica ions o quasi-
1D TMCs in mode n echnologies, Re . [158] demons a es how Nb3Se12I
can be used o high-capaci y op ical p ocessing and enc yp ion. This high-
ligh s he ich landscape o unc ional applica ions hese sys ems may en-
able.
Finally, he ou look o his Chap e is clea : he sys ema ic applica ion and
expansion o he amewo k p esen ed in Sec. 3.2in o de o build sound
knowledge ha allows he echnological applica ion o quasi-1D TMCs. Fo
example, his could be achie ed h ough he di ec - o-indi ec gap unning
and hence op ical esponse unning h ough doping o s ain.
4
2D, BULK TMDS & THEIR HETEROSTRUCTURES
Building on he insigh s gained om he s udy o quasi-1D TMCs, we now
shi ou ocus o TMDs and hei he e os uc u es, ma king a simul aneous
ansi ion in bo h dimensionali y and elec onic con igu a ion. These lay-
e ed ma e ials ha e a ac ed signi ican a en ion due o hei unable elec-
onic [24], magne ic [25], and op ical [26] p ope ies. They p o ide a e sa-
ile pla o m o in es iga ing quan um many-body phenomena, including
Kondo [30] and Mo [31] physics. Mo eo e , hei monolaye na u e, com-
bined wi h he abili y o o m he e os uc u es [32] h ough s acking o
wis ing, opens new a enues o explo ing phenomena ha a e di icul o
obse e in bulk ma e ials. No ably, in e laye coupling [27], moi é supe -
la ices [28], and s ong spin-o bi in e ac ions [29] a e in insic ea u es o
hese sys ems ha ende elec onic in e ac ions mo e complex and in e -
es ing.
We will begin by in oducing he subjec o his Chap e : TMDs and hei
he e os uc u es. We will p esen hei cha ac e is ic phenomenology ol-
lowing Re . [159] and highligh how hey di e om he quasi-1D TMCs
s udied in he p e ious Chap e . This includes an o e iew o hei c ys al
s uc u e, he ypes o CDW hey exhibi , and he mechanisms by which
hey can hos supe conduc i i y. Finally, we will b ie ly ou line he key
open ques ions ha his Chap e aims o add ess, and explain how he
subsequen Sec ions con ibu e o answe ing hem.
4.1 in oduc ion:ge o know mds
TMDs ha e a long his o y, da ing back o hei disco e y by Pauling in
he 1920s [23]. By he 1960s, app oxima ely 60 TMDs had been iden i ied,
wi h a leas 40 o hem exhibi ing a laye ed s uc u e. The i s epo ed
p oduc ion o MoS2monolaye suspensions da es back o 1986 [160]. The
ema kable success o g aphene- ela ed esea ch, beginning in 2004, led
o majo ad ancemen s in echniques o wo king wi h laye ed ma e ials,
pa ing he way o new s udies on TMDs and, mo e speci ically, on hei
ul a hin ilms.
In con as o quasi-1D TMCs, TMDs such as MoS2, WS2, NbSe2, and
TaS2a e cha ac e ized by a uni o m o mal oxida ion s a e o he ansi ion
me al, ypically M4+, which esul s in a egula dis ibu ion o d-elec ons
h oughou he la ice. This uni o mi y unde pins hei well-o de ed c ys al
97
4.1 in oduc ion:ge o know mds 98
Figu e 4.1: "Pe iodic Table" o known laye ed TMDs, o ganized based on he an-
si ion me al elemen in ol ed, summa izing hei exis ing s uc u al
phases (2H,1T o o he s). Adap ed om Re . [159]
s uc u es, which a e ypically composed o s acked laye s o edge-sha ing
MX6(M = ansi ion me al, X = chalcogen) uni s a anged in ei he igo-
nal p isma ic (2H phase) o oc ahed al (1T phase) coo dina ion [161] as we
see in he ollowing. The homogeneous elec onic en i onmen acili a es a
b oad ange o g ound s a es depending on he elec on coun and la ice
symme y: semiconduc ing beha io is obse ed in compounds like MoS2
and WS2, while me allici y and co ela ed phases such as CDWs and su-
pe conduc i i y eme ge in ma e ials like NbSe2and TaS2[133,159]. The
absence o mixed alence p e en s in insic cha ge modula ion, allowing
he elec onic p ope ies o be mo e di ec ly uned by ex e nal pa ame e s
such as doping, p essu e, o dimensional con inemen .
4.1.1Thei s uc u e
TMDs appea in a ple ho a o s uc u al phases, o igina ing om di e en
coo dina ion en i onmen s o he ansi ion me al a oms. The wo mos
common s uc u al phases a e cha ac e ized by igonal p isma ic (2H) o
oc ahed al (1T) coo dina ion o he me al a oms (see Fig. 4.4).
These s uc u al phases can be unde s ood in e ms o di e ences in he
s acking o de o he h ee a omic planes chalcogen-me al-chalcogen ha
o m he indi idual laye s o hese ma e ials. The 2H phase co esponds
o an ABA s acking, whe e chalcogen a oms in di e en a omic planes oc-
cupy he same posi ion (A) and a e aligned e ically. In con as , he 1T
phase is cha ac e ized by an ABC s acking o de . The he modynamically
s able phase, ei he 1T o 2H, depends on he speci ic combina ion o he
ansi ion me al ( om g oups IV, V, VII, IX, o X) and he chalcogen (S,Se,
o Te), see Fig. 4.1. Ne e heless, he o he phase can o en be ob ained as
a me as able con igu a ion.
4.1 in oduc ion:ge o know mds 99
The s uc u e o TMDs can be u he speci ied by he s acking con igu-
a ion o indi idual laye s in mul ilaye and bulk samples, as well as by
possible dis o ions ha educe pe iodici y. In he ollowing, we label he
wo-dimensional poly ypes wi h le e s Tand H, while he bulk poly ypes
a e deno ed as 1T, 2H, 4Hb, e c., whe e he numbe e e s o he numbe o
laye s in he uni cell.
4.1.2Thei CDWs.
G oup V laye ed dichalcogenides, in bo h he 2H and 1T phases o TaS2
and TaSe2, as well as in 2H-NbSe2, exhibi CDW o de [162,163]. Thei
ich and di e se phenomenology s ongly depends on he chemical compo-
si ion and c ys alline phase o he ma e ial. Fo ins ance, 2H-NbSe2unde -
goes a ansi ion a TCDW =33 K, leading o he o ma ion o a supe la ice
wi h app oxima ely a 3×3pe iodici y. Unlike o he CDW ma e ials, his
pe iodici y is almos independen o empe a u e and does no exhibi any
i s -o de ansi ion o commensu a e phases. The a omic displacemen is
e y small, on he o de o 0.05 Å, and he CDW has a ela i ely mode a e
impac on he physical p ope ies, causing only a sligh inc ease in esis i -
i y [164]. As he empe a u e is u he lowe ed, he ma e ial becomes su-
pe conduc ing [164]. By con as , 1T-TaSe2unde goes a sequence o incom-
mensu a e (T≈600 K), nea ly commensu a e (T≈350 K), and commensu-
a e (T≈180 K) ansi ions, he la e exhibi ing a √13 ×√13 pe iodici y
wi h a so-called S a o Da id (SoD) CDW, see Fig. 4.5. These ansi ions
a e accompanied by signi ican a omic displacemen s o Ta a oms, eaching
up o 0.24 Å, which s ongly a ec s he elec onic p ope ies. The ansi-
ion om he no mal s a e o he incommensu a e phase is second-o de
and is accompanied by a small jump in esis i i y. In con as , he nea ly
commensu a e and commensu a e ansi ions a e i s -o de and lead o a
subs an ial inc ease in esis i i y [165,166].
Following hese esul s on bulk TMDs, he s udy o CDW phases in mono-
laye s became a ocus o esea ch. DFT calcula ions ini ially sugges ed ha
monolaye H-NbSe2exhibi s a CDW phase wi h a di e en pe iodici y,
4×1, compa ed o he bulk s uc u e, 3×3×1, along wi h a g ea e elec-
onic ene gy gain and a highe ansi ion empe a u e, inc easing om
Tbulk
CDW =33 K o Tmono
CDW =145 K [167]. Howe e , u he expe imen al
esea ch ound no such enhancemen and ins ead epo ed a 3×3pe i-
odici y o he monolaye [168]. In he case o he isos uc u al H-TaSe2,
i s -p inciples calcula ions and expe imen al s udies sugges a 3×3CDW
s uc u e o bo h bulk and monolaye [169]. Se e al s udies ha e in es i-
ga ed he CDW phase in hin ilms o 1T-TaS2wi h hicknesses down o
a monolaye [170–172]. Howe e , he e olu ion o bo h he commensu a e
4.1 in oduc ion:ge o know mds 100
and nea ly commensu a e phases upon educing dimensionali y emains
unde deba e. In his manusc ip , we will adop he bulk √13 ×√13 CDW
s uc u e o 1T-TaS2.
Al hough Fe mi su ace nes ing has long been p oposed as a mechanism
o CDW o ma ion, i s applicabili y o TMDs and o he eal ma e ials e-
mains an open ques ion. I is now widely ecognized ha nes ing-d i en
CDW o ma ion is a highly idealized scena io, applicable p ima ily o s ic ly
one-dimensional sys ems o in excep ional cases [173,174]. In mo e com-
plex ma e ials, such as TMDs, he si ua ion is conside ably less s aigh -
o wa d: he lack o a clea ly de ined nes ing ec o , along wi h signi i-
can elec onic s uc u e econs uc ions induced by la ice dis o ions, ha e
led many esea che s o a gue ha Fe mi su ace nes ing plays only a mi-
no ole in he CDW o ma ion o hese sys ems. Ins ead, se e al s udies
p opose ha he obse ed dis o ions o igina e om s ongly aniso opic
elec on–phonon in e ac ions, whose ma ix elemen s a e signi ican ly en-
hanced a he CDW wa e ec o [167–169,175,176]. In he ollowing dis-
cussion, we do no engage in his deba e, bu a he ely on expe imen ally
epo ed o heo e ically p edic ed CDW s uc u es.
4.1.3Thei supe conduc i i y.
Al hough supe conduc i i y is no he cen al ocus o his Thesis, i eme ges
na u ally wi hin se e al o he sys ems unde in es iga ion and can be pa -
ially unde s ood by he ab ini io me hods employed. An example o his
appea s in Sec. 4.4, whe e we use DFT o unde s and how he sys em unde
s udy me allices and subsequen ly hos s supe conduc i i y. To p o ide he
necessa y backg ound o i s occasional appea ance h oughou he ex , a
b ie in oduc ion o he undamen als o supe conduc i i y is included in
Appendix C.9.
Is common o bulk TMDs ha exhibi a CDW s a e in hei phase diag am
o also display supe conduc i i y, wi h excep ions such as 2H-NbS2, which
only shows supe conduc i i y wi hou an accompanying CDW phase [177].
In TMDs, supe conduc i i y can be ei he in insic o induced h ough
chemical doping, elec os a ic doping, o applied p essu e. In 2H-NbSe2,
2H-TaS2, and 2H-TaSe2, supe conduc i i y coexis s wi h he CDW phase
a low empe a u es [178–180]. In con as , o 1T-phase TMDs such as 1T-
TaS2, he supe conduc ing s a e eme ges as he CDW o de mel s, as hap-
pens o he sys em s udied in Sec. 4.4. This ansi ion can be igge ed by
applying p essu e o h ough chemical doping, such as coppe in e cala-
ion.
Supe conduc i i y in TMDs pe sis s e en in he wo-dimensional limi .
4.1 in oduc ion:ge o know mds 101
T uly in insic supe conduc i i y has been epo ed in monolaye H-NbSe2
[168], wi h a c i ical empe a u e o T2D
C=3K, which is lowe han ha
o he bulk ma e ial, TBulk
C=7K. The in-plane c i ical ield equi ed o
supp ess he supe conduc ing s a e is an o de o magni ude la ge han
ha o he bulk ma e ial. This phenomenon is unde s ood o a ise om he
combined e ec s o educed dimensionali y and s ong SOC [181]. In mono-
laye Hphases, he spin spli ing a he Kand K′ alleys ac s as an e ec i e
ou -o -plane Zeeman ield. Due o spin- alley coupling, his spin spli ing
emains compa ible wi h Coope pai ing. As a esul , he supe conduc -
ing s a e in monolaye TMDs exhibi s an Ising-like spin pola iza ion in he
ou -o -plane di ec ion, making i signi ican ly mo e obus agains in-plane
magne ic ields.
4.1.4Open ques ions and ou line.
Once in oduced he key s uc u al and elec onic ea u es o TMDs and
hei he e os uc u es, we now u n o he cen al ques ions his Chap e
seeks o add ess. As we p esen ed, in con as o he quasi-1D TMCs o
he p e ious Chap e , TMDs o e a e sa ile pla o m whe e dimensional-
i y, in e laye coupling, and cha ge edis ibu ion can be enginee ed wi h
ema kable p ecision. Ye , despi e signi ican expe imen al and heo e ical
ad ances, undamen al aspec s o hei elec onic beha io , pa icula ly e-
ga ding cha ge ans e , Kondo and Mo phenomena along he eme gence
o me allici y and supe conduc i i y, emain insu icien ly unde s ood.
One open ques ion, o ins ance, conce ns he elec onic beha iou o T/H
he e os uc u es. As we will see in he ollowing, despi e bo h 1T and 2H
phases ha ing pa ially illed me al d-o bi al bands, hei low- empe a u e
monolaye p ope ies a e ma kedly di e en : while Hlaye s exhibi weak
CDWs and emain me allic and o en supe conduc ing [162], Tlaye s a e
unde s ood o be Mo insula o s in he √13 ×√13 CDW phase [182]. E en
mode a e Coulomb epulsion in he o bi als om Tlaye s is expec ed o
d i e he sys em in o a Mo insula ing s a e [182] and po en ially a spin-
liquid phase [183]. Howe e , when a Tlaye is placed on op o an Hlaye ,
cha ge ans e om he H o he Tlaye pa ially ills he la band (see Fig.
4.3 o see he Tmonolaye la band when in √13 ×√13 CDW phase), and
i emains unclea whe he he Mo s a e su i es unde hese condi ions.
Addi ionally, in his pa ially illed egime, he localized SoD momen s can
hyb idize wi h i ine an ca ie s in he Hlaye , gi ing ise o Kondo sc een-
ing. Bo h he supp ession o Mo insula ing beha io and he eme gence
o Kondo physics hus depend sensi i ely on he deg ee o cha ge ans e
be ween laye s.
Mo i a ed by his in e play, he i s pa o his Chap e , and he ame-
4.2 backg ound o ou heo e ial esea ch 102
wo k de eloped he ein, is de o ed o unde s anding cha ge ans e in
T/H he e os uc u es. In Sec. 4.3, we conduc a sys ema ic ab ini io in es-
iga ion o how a ious ac o s, such as in e laye spacing and chalcogen
composi ion, a ec cha ge ans e in TMD he e os uc u es. We ask: How
do hese ac o s in luence he edis ibu ion o elec onic cha ge, and how can his
be linked o he onse o supp ession o co ela ed elec onic phenomena? Ou goal
is o build a comp ehensi e pic u e o how s uc u al and chemical pa am-
e e s shape he elec onic en i onmen in MX2(M=Nb,Ta; X=S,Se) bilaye s,
and he eby gain insigh in o he eme gen low-ene gy beha io o Tand
Hmonolaye s in p oximi y.
Meanwhile, in Sec. 4.4, we combine DFT calcula ions wi h expe imen al
esul s om ou collabo a o s in he g oup o M. Ugeda o add ess ano he
open ques ion: Why is bulk 1T-TaSSe me allic? While his ques ion migh
seem a bi naï e, i s all o he mo e in e es ing: p e ious expe imen al s ud-
ies based on ARPES and STM claim bulk 1T−TaX2(X=S,Se) o be Mo
insula ing [59,60]. Howe e , mo e ecen in es iga ions like Re . [39] poin
ou dime iza ion as he main mechanism o he insula ing beha iou . In
Sec. 4.4we iden i y he pa hway by which an a p io i insula ing sys em,
closely ela ed o he Mo -insula ing 1T-TaS2, becomes me allic and supe -
conduc ing. Th ough his combined heo e ical and expe imen al app oach,
we iden i y s acking diso de in he CDW phase as he key mechanism,
p o iding a mic oscopic unde s anding o me alliza ion in bulk 1T-phase
TMDs.
4.2 backg ound o ou heo e ial esea ch
Building on he insigh s collec ed om he s udy o TMDs’ phenomenol-
ogy, we now p esen ou heo e ical wo k on cha ge ans e in T/H he -
e os uc u es. We i s mo i a e ou esea ch by discussing he physical
phenomena ha hese sys ems can hos , namely, Mo and Kondo physics.
To ha end we conduc a ho ough DFT s udy o de e mine he condi ions
unde which each he e os uc u e could ealize hese phases. This s udy
aims o p o ide a solid heo e ical ounda ion o u u e expe imen al and
heo e ical in es iga ions.
Despi e bo h 1T and 2H phases ha ing pa ially illed me al d-o bi al bands,
hei low- empe a u e monolaye p ope ies a e ma kedly di e en : while
Hlaye s exhibi weak CDWs and emain me allic and o en supe conduc -
ing [162], Tlaye s a e unde s ood o be Mo insula o s [182]. This beha io
is a ibu ed o he p e iously discussed √13 ×√13 SoD CDW econs uc-
ion [184], which gaps ou mos o he Fe mi su ace, lea ing a hal - illed
la band a he Fe mi le el de i ed om an isola ed o bi al cen e ed a
he SoD. E en mode a e Coulomb epulsion in hese o bi als is expec ed o
4.2 backg ound o ou heo e ial esea ch 103
d i e he sys em in o a Mo insula ing s a e [182] and po en ially a spin-
liquid phase [183]. Howe e , he ealiza ion o such s a es in he bulk 1T
poly ype, composed o s acked Tlaye s, is complica ed by in e laye un-
neling and he mul iple possible CDW s acking pa e ns [39]. The ecen
success ul syn hesis o Tmonolaye s [40] has p o ided s onge e idence
suppo ing he Mo insula o scena io.
TMD he e os uc u es ha al e na e Tand Hlaye s o e an in e es ing
pla o m o es he Mo insula o hypo hesis. In hese s uc u es, he local-
ized SoD momen s in he Tlaye couple o he me allic elec ons in he H
laye , na u ally ealizing a CDW-induced Kondo la ice. Recen s udies on
syn hesized T/H bilaye s ha e indeed e ealed p ominen ze o-bias peaks
in TaSe2[40,41], TaS2[42,43], and NbSe2[44,45], which ha e been in-
e p e ed as signa u es o he Kondo e ec . Beyond isola ed bilaye s, wo
na u ally occu ing bulk poly ypes o med by al e na ing Tand Hlaye s
also exis : 4Hb[47,48], wi h an in e sion-symme ic s acking o T/H bi-
laye s, and 6R [49,50], wi h an in e sion-b eaking hombohed al s acking.
In e es in hese compounds has ecen ly been enewed, as hey no only
exhibi simila Kondo e ec s [51,52], bu also shows unexpec ed signa-
u es o uncon en ional supe conduc i i y [53] (speci ically in 4Hb-TaS2),
including spon aneous ime- e e sal symme y b eaking a Tc[54], spon a-
neous o ex o ma ion in he supe conduc ing s a e [55], supe conduc ing
edge modes [56], and anspo e idence o a wo-componen o de pa am-
e e [57,58].
The in e p e a ion o hese expe imen s, as well as he alidi y o he Mo
insula o pic u e, c i ically depends on wo key p ope ies o he T/H in-
e ace, which emain unde deba e: he in e laye hyb idiza ion Vand he
cha ge ans e ∆C = (CH−CT)/2, whe e CH=CTin acuum, wi h he
ac o 1/2 accoun ing o double-coun ing. The Tlaye can only emain a
Mo insula o i he la band emains nea ly hal - illed. Howe e , since he
wo k unc ion o he Hlaye is la ge han ha o he Tlaye , some cha ge
ans e om T o H is expec ed [185–187]. Simila ly, he Kondo e ec can
only su i e wi hin a limi ed ange o cha ge ans e and equi es a c i ical
hyb idiza ion V > Vc. Supe conduc i i y is also expec ed o beha e di e -
en ly depending on whe he he Tlaye con ibu es magne ic momen s. A
ecen s udy has p oposed a scena io in which he Tlaye beha es as a
doped Mo insula o wi h negligible hyb idiza ion V[46]. Impo an ly, he
ole o in e laye cha ge ans e in bulk 4Hbcompounds has no ye been
explo ed, no has he po en ial a iabili y among di e en membe s o his
amily, such as TaS2, TaSe2, o NbSe2.
4.2 backg ound o ou heo e ial esea ch 104
4.2.1A b ie no e on Mo & Kondo physics.
This manusc ip is p ima ily ocused on single-pa icle physics. Ne e he-
less, a single-pa icle app oach, mainly h ough ab ini io me hods, can p o-
ide aluable insigh s in o he collec i e many-body beha io o he sys-
ems unde s udy. This is pa icula ly ue o he TMD he e os uc u es
ha we examine ex ensi ely in his Chap e . As men ioned in he p e i-
ous Sec ion, he in e p e a ion o expe imen s and he de e mina ion o he
many-body na u e o TMDs c i ically depend on in e laye hyb idiza ion
and cha ge ans e be ween laye s. These key pa ame e s can be ex ac ed
om a DFT s udy, which in u n helps illumina e he physical unde s and-
ing o hese sys ems, commonly ega ded as Mo insula o s o Kondo la -
ices. In he ollowing Sec ions, we p o ide a b ie in oduc ion o hese
collec i e phenomena o es ablish he g oundwo k o ou esea ch.
4.2.1.1Mo ansi ion
As an icipa ed in he In oduc ion, educing TaX2wi h X=S,Se TMDs o
hei monolaye s uc u es, T-TaX2, whe e co ela ion e ec s can be mo e
p onounced, aises undamen al ques ions abou he na u e o elec onic
co ela ions in hese sys ems, ques ions ha we will explo e in he ollow-
ing pages. To es ablish a heo e ical ounda ion, we begin by in oducing
he me al- o-insula o Mo ansi ion in a basic manne .
Condensed ma e physics deals wi h sys ems composed o an ex emely
la ge numbe o elemen a y cons i uen s, ypically on he o de o 1023.
This esul s in a as numbe o deg ees o eedom, making hei unde -
s anding and desc ip ion challenging wi h a pu ely educ ionis app oach.
New physical p inciples a e hus equi ed o desc ibe such sys ems as
a whole [188]. Eme gen collec i e phenomena, a ising om he in e ac-
ions among mic oscopic cons i uen s, a e ubiqui ous. These phenomena
canno be unde s ood me ely by conside ing indi idual cons i uen s in iso-
la ion; a he , hey eme ge om he coope a i e beha io o he sys em as a
whole. No able examples include supe conduc i i y and magne ism, bo h
o which a e d i en by spon aneous symme y b eaking1. In mos cases,
in e ac ing sys ems mus be app oxima ed using a minimal desc ip ion in
e ms o weakly in e ac ing quasipa icles. Examples include Bogoliubo
quasipa icles in he Ba deen-Coope -Sch ie e (BCS) model o supe con-
duc i i y, phonons in la ices wi h b oken ansla ional symme y, and spin
wa es in magne ic sys ems [189]. In his discussion, we will ollow Re .
[190] o in oduce and ge a big pic u e on Mo physics and how can i
appea on T/H TMD he e os uc u es.
1I is conside ed spon aneous as i eme ges om he in insic in e ac ions o he sys em
wi hou equi ing ex e nal ields o pe u ba ions.
4.3 cha ge ans e in T/H he e os uc u es o monolaye mds.111
as a cha ac e is ic esis i i y up u n a low empe a u es. This e ec is in-
he en ly non-pe u ba i e and is bes unde s ood wi hin he amewo k
o eno maliza ion g oup heo y, which shows ha he e ec i e coupling
be ween he localized momen and conduc ion elec ons g ows s onge
as empe a u e dec eases. Beyond i s undamen al implica ions in s ongly
co ela ed elec on sys ems, Kondo physics plays a c ucial ole in a wide
a ie y o ma e ials, including hea y e mion compounds, quan um do s,
and enginee ed he e os uc u es, whe e localized spins in e ac wi h con-
duc ion elec ons. This las example would be he case on T/H he e os uc-
u es.
In he con ex o T/H TMD he e os uc u es, he po en ial eme gence o
Kondo sc eening depends on he balance be ween in e laye cha ge ans-
e and hyb idiza ion, which de e mines whe he he Tlaye e ains a local-
ized momen o becomes me allic. All necessa y ing edien s o he Kondo
e ec a e p esen : a nea ly la band in he monolaye Tpoly ypes (see Fig.
4.3), associa ed wi h he √13 ×√13 CDW s a e, could ac as an e ec i e
spin-1/2 localized magne ic momen i i emains hal - illed. Meanwhile,
he me allic monolaye Hpoly ype p o ides a conduc ion elec on sea ca-
pable o in e ac ing wi h a localized momen in he Tlaye , po en ially lead-
ing o a Kondo s a e. Cha ge ans e and in e laye hyb idiza ion a e hus
key pa ame e s de e mining he egime o T/H he e os uc u es. A ho -
ough sys ema ic ab-ini io cha ge ans e s udy will be he guiding heme
o he ollowing pages.
4.3 cha ge ans e in T/H he e os uc u es o monolaye
mds.
In his Sec ion, we sys ema ically in es iga e cha ge ans e ac oss all hese
compounds, analyzing i s co ela ion wi h wo k unc ion misma ch, Van
de Waals co ec ions, Hubba d’s Upa ame e , and in e laye spacing. Un-
de s anding cha ge ans e is c ucial o de e mining he elec onic na u e
o hese sys ems, as i di ec ly in luences whe he he Tlaye e ains i s
Mo -insula ing cha ac e o ansi ions in o a Kondo la ice by in e ac ing
wi h conduc ion elec ons om he Hlaye . Ou calcula ions e eal ha he
impac o Uand Van de Waals e ec s is minimal, while cha ge ans e
is p edominan ly go e ned by he in e laye spacing and he misma ch in
wo k unc ions. Mo eo e , we iden i y a gene al end whe e Se-based com-
pounds exhibi lowe cha ge ans e han hei S-based coun e pa s, and
4Hbbulk poly ypes display s onge cha ge ans e han isola ed bilaye s.
These indings p o ide a solid ounda ion o unde s anding he deg ee o
cha ge ans e in T/H he e os uc u es and i s implica ions o eme gen
many-body physics, o e ing a p ac ical cookbook o p edic ing which sys-
ems exhibi g ea e o lowe cha ge ans e unde speci ic condi ions.
4.3 cha ge ans e in T/H he e os uc u es o monolaye mds.112
°M K °
°1.0
°0.5
0.0
0.5
1.0
E-E
(eV)
NbS2
°M K °
°1.0
°0.5
0.0
0.5
1.0
E-E
(eV)
NbSe2
°M K °
°1.0
°0.5
0.0
0.5
1.0
E-E
(eV)
TaS2
°M K °
°1.0
°0.5
0.0
0.5
1.0
E-E
(eV)
TaSe2
Figu e 4.3: Elec onic bands uc u es om he Tmonolaye s o he sys ems consid-
e ed in he √13 ×√13 CDW.
Ea ly wo ks [185–187] al eady an icipa ed ha cha ge ans e om he
T o he Hlaye s mus be p esen in bulk 4HbTMDs. T/H bilaye and bulk
4Hb- MX2s uc u es a e shown in Fig 4.4. A high empe a u es whe e he
CDW in he Tlaye s is incommensu a e, he change in CDW wa e ec o
compa ed o bulk 1T poly ypes was used o es ima e a ans e o 0.12 e−
pe o mula uni [185] (1.56 e−pe SoD) in 4Hb-TaS2, and a simila es ima e
leads o 1.20 e−pe SoD o TaSe2. Simila heo e ical es ima es [187] o
4Hb-TaS2simila ly anged be ween 1.04 o 1.43 e−pe SoD. Cha ge ans-
e o his magni ude was also epo ed o be consis en wi h changes in
he op ical conduc i i y in bo h 4Hband 6R poly ypes [186]. Mo e ecen
ARPES expe imen s es ima ed 0.92 e−pe SoD in 4Hb-TaS2[201]. All hese
ea ly es ima es a e hus consis en wi h a nea ly emp y la band.
Recen ab-ini io calcula ions in he high empe a u e s a e wi hou CDW [202,
203] also sugges cha ge ans e om T o H, bu gi en he s ong band
econs uc ion due o he CDW, i is impo an o pe o m hese calcula-
ions in he CDW s a e8. Such calcula ions [52,56] o a T/H bilaye o TaS2
8In he ollowing, in DFT calcula ions we conside Tlaye √13 ×√13 CDW bu no he
3×3 H laye CDW because o he s eep compu a ional cos i would imply. Mo eo e , i
4.3 cha ge ans e in T/H he e os uc u es o monolaye mds.113
M
X
Hb - MX
4
2
T/H - MX
2
T
H
T’
H’
T
H
Figu e 4.4: Bulk 4Hb(le ) and bilaye T/H ( igh ) MX2(M=Nb,Ta; X=S,Se) s uc-
u es conside ed in his Chap e .
s ill epo a ully emp y la band, while a bilaye Ton monolaye H e-
po ed 0.31 e−pe SoD cell [205]. Fo TaSe2, a alue o 0.32 e−pe SoD cell
was epo ed [41], while o NbSe2 0.17 e−was calcula ed [204]. A ecen
s udy has emphasized he impo ance o he s acking dis ance on cha ge
ans e [46], e ealing ha ∆C anges om 0.4 o 1in TaS2as he in e laye
dis ance goes om 7 o 5.8Å.
Gi en such a iabil y, and he ac ha isola ed bilaye s on subs a es may
no s ack wi h he same in e laye dis ance as bulk 4Hbcompounds, i is
impo an o s udy he dis ance dependence in de ail o TaSe2and NbSe2.
Di e ences may be expec ed because he la band in he Se compounds
is signi ican ly close o he CDW alence bands compa ed o he S com-
pounds, his is shown in Fig. 4.3. Fo NbSe2one expe imen has claimed
[45] a cha ge ans e wi h an opposi e sign o ha o he Ta compounds. I
is also impo an o ake in o accoun he de ails o he di e en ab-ini io
calcula ions done o he Ta compounds [206–212] , o example because he
exac posi ion o he la band wi hin he CDW gap is known o depend on
he unc ional used [213], which can in luence cha ge ans e . Including
he Hubba d in e ac ion and explici ly accoun ing o magne ic s a es wi h
spin-spli bands [213–217] can simila ly a ec he cha ge ans e .
is a common p ac ice since i is no expec ed o ha e a huge impac in cha ge ans e [41,
46,204].
4.3 cha ge ans e in T/H he e os uc u es o monolaye mds.114
Finally, no e in he con ex o co ela ed sys ems he e m cha ge ans e
is o en used o emphasize he dis inc ion be ween Mo and cha ge ans-
e insula o s [218]. In ha case, he e m e e s o cha ge ans e be ween
co ela ed d-de i ed bands and he dispe si e p-de i ed bands o he same
compound. In TMDs his phenomenon may also be ele an a leas o
some compounds like 1T-NbSe2o 1T-TaSexTe1−xwhe e he la band may
o e lap wi h he p-de i ed s a es [44,213,219]. In ou wo k, unless speci-
ied o he wise, cha ge ans e will a he e e o in e laye cha ge ans e
be ween he Tand he Hlaye s.
4.3.1Ab-ini io me hods
The aim o his chpa e is o p o ide a sys ema ic s udy o he in e laye
cha ge ans e be ween Tand HMX2laye s. The wo k low used o ca y
ou said s udy is as ollows:
1. Single laye in-plane s uc u e elaxa ion o Hand Tlaye s.
2. Single laye wo k unc ion calcula ion o Hand Tlaye s.
3. T-H bilaye elaxa ion, i s elaxing in he ˆzdi ec ion ollowed by a
subsequen in-plane elaxa ion.
4. Cha ge ans e calcula ion o bilaye s. In his s ep we explo e di e -
en pa ame e s as Hubba d U, Van de Waals co ec ions and in e -
laye dis ance dependence.
5.4Hbcha ge ans e calcula ion using s ep 2s uc u es and expe i-
men al dis ances.
The pu pose o his wo k low is o iden i y ends be ween di e en ac o s
a ec ing ab-ini io calcula ions and expe imen al measu emen s, aiming o
unde s and he o e all beha iou o cha ge ans e unde di e en condi-
ions.
All calcula ions we e pe o med using Vienna Ab ini io Simula ion Pack-
age (VASP) [67,68] .6.2.1. wi h p ojec o -augmen ed wa e pseudopo en-
ials wi hin he Pe dew Bu ke E nze ho pa ame iza ion [126]. Fo s ep 1
and 3, he elaxa ion was conduc ed by using he conjuga e-g adien algo-
i hm as implemen ed in VASP, keeping he cell shape and olume ixed
while le ing a omic posi ions elax. Fo s eps 1−4, calcula ions we e ound
o be well con e ged wi h a 480 eV kine ic cu o and a gamma-cen e ed
15 ×15 ×1 k−mesh. Meanwhile, o s ep 5, he sel -consis en calcula ions
we e ound o be well con e ged wi h a 480 eV kine ic cu o and a gamma-
cen e ed 13 ×13 ×3 k−mesh. In s ep 4, when Van de Waals co ec ions
4.3 cha ge ans e in T/H he e os uc u es o monolaye mds.115
C
B
A
Figu e 4.5: La ice s uc u e o he SoD CDW dis o ion o he Tlaye s. A, B, C
inequi alen me al si es a e ma ked in yellow, g een and blue espec-
i ely. Black a ows show hei displacemen s while a do ed line ma ks
he CDW uni cell.
we e conside ed, DFT-D3me hod [220] wi h ze o damping was used. Also
in s ep 4, in o de o s udy he e ec o Coulomb epulsion, he DFT+U
o a ionally in a ian app oach [221] was ollowed by se ing di e en e -
ec i e on-si e UCoulomb in e ac ions in M’s d−o bi als wi h J=0. These
DFT+U calcula ions a e he only collinea spin-pola ized ones. The ini ial
magne iza ion was se o 1.4µB o he cen al Ta/Nb a om (A a om, see Sec.
4.3.2), and ze o o he emaining a oms. This choice is based on p io s ud-
ies sugges ing ha he p edominan magne ic momen is concen a ed a
he cen e o he SoD [208], and ha he o al magne ic momen is ypically
a ound 1µB[214,216]. The ini ial alue o magne iza ion is aken sligh ly
la ge han he expec ed esul as his is expec ed o imp o e con e gence
[222]. A om-p ojec ed band s uc u es we e ob ained using PyP oca [223]
package o Py hon.
4.3.2Cha ge densi y wa e dis o ions
The √13 ×√13 SoD CDW s uc u e is shown in Fig. 4.5. The e a e h ee
ypes o symme y equi alen M si es labeled as A, B and C he ea e , wi h
mul iplici ies 1,6, and 6 espec i ely. The s uc u e is pa ame e ized by
h ee independen displacemen s uiwi h i=A,B,C, shown in Fig. 4.5
(|uA|=0by symme y). In able 3we p esen he displacemen s o each
equi alen me al si e in each T-MX2 om he bilaye s conside ed. In Ap-
pendix C.6we p esen a g aphic depic ion o hese displacemen s om he
Tlaye o he bilaye o he ou compounds (Fig. C.5) along a b ie no e
on how we ob ained he CDW posi ions.
4.3 cha ge ans e in T/H he e os uc u es o monolaye mds.116
4.3.3Wo k unc ion analysis
We de ine he wo k unc ion as he absolu e Fe mi le el wi h espec o ac-
uum. Di e en wo k unc ions be ween wo s uc u es indica e misaligned
Fe mi le els and can be used o a quali a i e es ima e o cha ge ans e .
To compu e he wo k unc ions, we pe o m a sel -consis en calcula ion o
he monolaye in a uni cell wi h ≈20 Å o acuum in he no mal di ec ion.
F om his calcula ion we ex ac he local po en ial V( )and de ine he wo k
unc ion as
W=V ac −EF, (4.15)
whe e V ac is he alue o V( )in accuum. We p esen he wo k unc ions
o bo h Tand Hpoly ypes o all TMDs conside ed in able 3.Hwo k unc-
ion is g ea e han Two k unc ion o all TMDs which an icipa es ha he
cha ge ans e will occu om T o Hlaye . In Table 3we can al eady see
some ends: M=Ta compounds ha e o e all smalle wo k unc ion han
M=Nb and so happens wi h X=Se when compa ed wi h X=S ones. In Sec-
ion 4.3.4we will see how his a ec he cha ge ans e .
The wo k unc ion esul s in Table 3a e consis en wi h Re . [205], which
epo ed wo k unc ion alues o W=5.35 eV o T-TaS2in he CDW s a e,
and W=6.07 eV o H-TaS2in he 3×3CDW s a e. He e we calcula ed
W=5.19 eV o T-TaS2in he CDW s a e, and W=5.57 eV o H-TaS2
wi hou CDW. We op ed no o include he 3×3CDW s a e in H-TaS2
laye s in ou calcula ions due o he excessi e compu a ional cos o a uni
cell commensu a e wi h bo h √13 ×√13 and 3×3CDW s a es. Howe e ,
ou analysis o he cha ge ans e as unc ion o wo k unc ion di e ences
sugges s his is a good app oxima ion.
4.3.4Cha ge ans e
Cha ge ans e is he cen al esul o his wo k. To calcula e he cha ge
ans e be ween he di e en cons i uen s o a he e os uc u e we ollowed
he me hod p esen ed in Re s.[205,224]. This me hod is based on compu -
ing he cha ge densi y o he ull s uc u e wi h espec o ha o a hy-
po he ical e e ence s uc u e buil om he calcula ed cha ge densi ies o
he isola ed cons i uen s, posi ioned in he places hey would occupy in he
ull s uc u e. The choice o his me hod elays on wo main pila s: i s ly, i
only needs he main ou pu om DFT calcula ions (i.e. cha ge densi y) wi h
small p ocessing, le ing ou sys ema ic app oach o be pe o med wi hou
calcula ion o e head. And secondly, as showed in Re s. [205,224], i can
p o ide a quan i a i e measu emen o cha ge ans e . Mo e explici ly, i
we conside ˆzas he s acking di ec ion, we can ob ain he plane-a e aged
cha ge densi y om he sel -consis en calcula ions ρall(z)and ρi(z)wi h
4.3 cha ge ans e in T/H he e os uc u es o monolaye mds.117
NbSe2NbS2TaSe2TaS2
|uA|(Å) 0.00 0.00 0.00 0.00
|uB|(Å) 0.26 0.21 0.26 0.20
|uC|(Å) 0.32 0.28 0.31 0.25
d(Å) 7.37 6.89 7.51 7.10
WT(eV) 5.25 5.47 5.02 5.19
WH(eV) 5.57 6.13 5.45 5.96
∆WF (eV) 0.32 0.66 0.43 0.77
CT (e) 0.12 0.23 0.14 0.26
Table 3: Displacemen s o each inequi alen me allic posi ion uiwi h i=A,B,C, in e laye dis ance d, wo k unc ions WTand WH o T
and Halong ∆WF and cha ge ans e CT o all ou compounds.
4.3 cha ge ans e in T/H he e os uc u es o monolaye mds.118
0.00
2.50
Ω(e/˚
A)
1T 1H
1T/H - TaS2
-0.01
0.00
0.01
Ωdi (e/˚
A)
0 5 10 15 20 25
z(˚
A)
0.00
0.20
q (e)
1.00
2.00
3.00
Ωall (e/˚
A)
1T 1H 1T’ 1H’
4Hb-TaS
2
0.00
0.03
Ωdi (e/˚
A)
0 5 10 15 20
z(˚
A)
0.00
0.50
q (e)
T H
Figu e 4.6: Elec onic densi y ( op), elec onic densi y di e ence (mid) and o al
cha ge change (bo om) o T/H TaS2bilaye . The ze o o elec onic
densi y di e ence is ma ked wi h a e ical dashed line.
i=1,2,...,N, whe e Nis he numbe o componen , and ob ain he o e all
cha ge densi y di e ence as
ρdi (z) = ρall(z) −
N
X
i=1
ρi(z), (4.16)
In eg a ing ρdi (z)we can ob ain he o al cha ge di e ence in a sec ion
(z1,z2)as
qdi (z1,z2) = Zz2
z1
ρdi (z)dz. (4.17)
De e mining app op ia e alues o z1and z2is s aigh o wa d o bilay-
e s: as depic ed in Fig. 4.6,z1may be posi ioned anywhe e in acuum,
while z2=2ma ks he poin whe e ρdi (z2) = 0nea es o he in e ace.
q(z1,z2) ep esen s he o al cha ge di e ence o he i s componen and
he o e all absolu e cha ge ans e be ween componen s. Howe e , choos-
ing zican be challenging when N > 2 o o pe iodic sys ems, such as
he 4Hbs uc u e. A p ac ical app oach is o se z1as he poin whe e
ρdi (z1) = 0nea es o he in e ace wi h he p eceding componen , and z2
as he poin whe e ρdi (z2) = 0nea es o he in e ace wi h he subsequen
componen . Fo ins ance, in Fig. 4.7, he o al cha ge di e ence o he T
laye in 4Hbis ob ained by in eg a ing om z1=0Å o z2=6.40 Å (whe e
he e ical and ho izon al dashed lines i s in e sec in he second g aph);
4.3 cha ge ans e in T/H he e os uc u es o monolaye mds.119
1.00
2.00
3.00
ρal l (e/˚
A)
T H T’ H’
4Hb- TaS2
0.00
0.03
ρdi (e/˚
A)
0 5 10 15 20
z(˚
A)
0.00
0.50
q (e)
Figu e 4.7: Elec onic densi y ( op), elec onic densi y di e ence (mid) and o al
cha ge change (bo om) o 4HbTaS2. Ze os o elec onic densi y di -
e ence and hus candida es o z1/z2a e ma ked wi h e ical dashed
lines.
o H, in eg a ion anges om z′
1=z2=6.40 Å o z′
2=11.95 Å , and so
o h. These in eg a ion limi s co espond o he maximum and minimum
alues o q(0,z)as illus a ed in he hi d g aph.
In Table 3we p esen he cha ge ans e esul s o he elaxed in e laye
dis ances. These dis ances a e 6.89Å o NbS2,7.37Å o NbSe2,7.10Å o
TaS2and 7.51Å o TaSe2. Those alues a e plo ed in Fig. 4.8, whe e he
p e iously men ioned CT ∝∆WF end is clea . F om hese esul s, we
can de ec some ends: ab-ini o calcula ions p edic Ta compounds o ha e
g ea e cha ge ans e han Nb compounds, and S compounds ha e g ea e
cha ge ans e han Se compounds.
4.3.5Van de Waals e ec
In his Sec ion, we compa e he calcula ion o he cha ge ans e wi h
and wi hou Van de Waals co ec ions o he T/H s uc u es ob ained
in Sec. 4.3.4. To do so we inco po a ed Van de Waals co ec ions in o he
elec onic sel -consis en calcula ions and epea ed he cha ge ans e cal-
cula ions in Sec. 4.3.4. Ou indings indica e ha hese co ec ions ha e a
negligible in luence on cha ge ans e , ypically on he o de o < 10−3.
While Van de Waals co ec ions he e o e do no a ec cha ge ans e
di ec ly, he could do so indi ec ly i we pe o med a new elaxa ion o he
4.3 cha ge ans e in T/H he e os uc u es o monolaye mds.120
0.30 0.40 0.50 0.60 0.70 0.80
∆ Wo k unc ion (eV)
0.10
0.12
0.15
0.18
0.20
0.23
0.25
0.28
Cha ge T ans e (e)
NbSe2
TaSe2
NbS2
TaS2
Figu e 4.8: Cha ge ans e as a unc ion o wo k unc ion di e ence be ween laye s
in T/H bilaye .
s uc u e in he p esence o such co ec ions, since he in e laye dis ance
could change upon elaxa ion. Since ou main in e es in his wo k is o
es ablish ela i e ends in cha ge ans e , we ha e a he op ed o s udy
cha ge ans e as a unc ion o in e laye dis ance wi hou a emp ing o
calcula e p ecisely i s equilib ium alue, as his is a mo e complex p oblem
ha depends on bo h calcula ional de ails and expe imen al condi ions.
4.3.6U dependence
Following he same logic as in he p e ious Sec ion, we now conside he
e ec o he Hubba d in e ac ion U, only a he le el o elec onic sel -
consis en calcula ions keeping he s uc u e ixed 9. We conside ed only
TaS2as an example. The main e ec o he Hubba d in e ac ion is o mag-
ne ize he la band nea he Fe mi le el (see Fig. 4.3), p oducing a spin
spli ing and pushing one o he spin pola iza ions abo e he Fe mi le el.
Since a la ge magne iza ion is p oduced when he lowe spin-spli band is
close o hal - illing, inc easing Ugene ally leads o a educ ion in cha ge
9To e i y ou decision o disca ding Uin he s uc u al elaxa ion, we conduc ed a es
elaxa ion using he same p ocedu es desc ibed in 4.3 o U=2.0eV in TaS2. We ound
ha he e ec o including Uon he elaxa ion only changes ionic posi ions on he o de o
10−4−10−5Å. These esul s a e consis en wi h he indings o L. C ippa e al. in Re .[46],
which sugges ha he o e all in luence o Uin DFT calcula ions o hese compounds is
small.
4.4 expe imen al ealiza ion:me allici y h ough andom s acking.127
cha ge ans e acc oss he amily o T/H s uc u es, which will lead o a
deepe unde s anding o he uncon en ional magne ic and supe conduc -
ing p ope ies in his amily o ma e ials.
4.4 expe imen al ealiza ion:me allici y h ough andom
s acking.
In he p e ious Sec ion, we explo ed cha ge ans e and i s po en ial ole
in de e mining he collec i e elec onic s a e o T/H he e os uc u es. In
wha ollows, we p esen DFT calcula ions ha suppo ed ou expe imen al
collabo a o s om he g oup o M. Ugeda in cha ac e izing and unco e ing
he mic oscopic o igin o supe conduc i i y in bulk 1T-TaSSe (1T-TaS2−xSex
wi h x=1). To ha end, we s udied how an a p io i insula ing sys em
becomes me allic and subsequen ly hos s supe conduc i i y.
4.4.1In oduc ion & Backg ound
Oc ahed ally coo dina ed 1T bulk poly ypes, such as 1T-TaS2and 1T-TiSe2,
exhibi simila elec onic phase diag ams, like he one shown in Fig. 4.13 (b).
No ably, nei he o hem is supe conduc ing in i s p is ine o m; howe e ,
supe conduc i i y can be induced in hem h ough doping [170,230], p es-
su e [231], chemical subs i u ion, o in e cala ion [232,233]. In bo h com-
pounds, hese ex e nal pe u ba ions d i e he sys em h ough a sequence
o phase ansi ions: om a CCDW s a e o an ICDW s a e, p eceding he
eme gence o supe conduc i i y. The ansi ion o an ICDW phase esul s
in he o ma ion o CCDW domains sepa a ed by sha p, in e connec ed do-
main walls, which we e hypo hesized o be he d i ing o ce behind he
onse o supe conduc i i y [230,231]. Howe e , despi e hese insigh s, ex-
ac knowledge on how his ansi ion o a ICDW leads o me alliza ion and
he subsequen eme gence o supe conduc i i y emains lacking o TaS2.
The eme gence o supe conduc i i y upon ex e nal pe u ba ion is pa icu-
la ly in e es ing in he case o 1T-TaS2, which exhibi s an insula ing g ound
s a e a low empe a u es, cha ac e ized by a √13 ×√13 CDW. The o igin
o his insula ing beha io has been la gely a ibu ed o a possible Mo
s a e a ising om s ong elec onic co ela ions wi hin he CDW [234–236].
Howe e , mo e ecen in e p e a ions sugges ha he insula ing gap o ig-
ina es om in e laye CDW coupling [39,237–239]. The applica ion o he
a o emen ioned pe u ba ions ypically leads o me alliza ion o he ma e-
ial, which subsequen ly enables he eme gence o supe conduc i i y, wi h
op imal c i ical empe a u es TC anging om 1 o 6K [170,240] as shown
in he phase diag am om Fig. 4.13 (b).
In he ollowing, we iden i y he andomiza ion o he CDW s acking o -
4.4 expe imen al ealiza ion:me allici y h ough andom s acking.128
CCDW
(Insula ing)
ICDW
(Me allic)
SC
No mal me al
X
Tempe a u e
a
b
cde
Uppe mos 1T laye Second 1T laye
1.7 Å
01.7 Å
0
2.4 Å
0
10 nm
2 nm 2 nm
CCDW
(Insula ing)
ICDW
(Me allic)
SC
No mal me al
X
Tempe a u e
a
b
cde
Uppe mos 1T laye Second 1T laye
1.7 Å
01.7 Å
0
2.4 Å
0
10 nm
2 nm 2 nm
a b
Figu e 4.13: Gene ic phase diag am o 1T −TMD me als whe e x a iable in he ˆx
axis e e s o Se con en upon Ssubs i u ion.
de as he main d i e o he me allic beha io in 1T-TaSSe, based on a
combined analysis o STM/STS expe imen s pe o med by ou collabo a-
o s and ou own DFT calcula ions. This in e p e a ion is mo i a ed by
p e ious s udies on isos uc u al compounds [216,239,241], whe e s ack-
ing diso de has been linked o he supp ession o insula ing beha io . To
in es iga e his mechanism, STM/STS measu emen s we e pe o med a
di e en spa ial loca ions, e ealing local a ia ions in he elec onic s uc-
u e. By compa ing hese measu emen s wi h DFT simula ions o dis inc
s acking con igu a ions, we we e able o associa e each local spec um wi h
a speci ic s acking o de , p o iding a mic oscopic unde s anding o he
me allici y in e ms o s acking diso de .
4.4.2dI/dV spec oscopy
The sys em unde s udy, 1T-TaSSe, is a ela ed compound o 1T-TaS2. I
becomes me allic10 upon seleniza ion [240,242]: a p ocess consis sing in an
iso alen subs i u ion o Sby Se. A g adual seleniza ion o 1T-TaS2−xSex
leads o he mel ing o he CCDW o x > 0.8, a o ing he ICDW ollowed
by he eme gence o supe conduc i i y in he ange 0.9<x<1.6, as shown
in Fig. 4.13 (b).
Th oughou his Thesis, we e e o he ICCDW phase obse ed in 1T-TaSSe
as a mosaic phase. This e minology cap u es he spa ially inhomogeneous
cha ac e o he cha ge-o de ed s a e, in which locally commensu a e SoD
clus e s coexis wi h domain bounda ies and de ec s, o ming a pa e n
10 In he p esen ed e e ences, said compound becomes me alic, hos s and op imizes supe -
conduc i i y upon seleniza ion.
4.4 expe imen al ealiza ion:me allici y h ough andom s acking.129
eminiscen o a diso de ed mosaic. This ICCDW phase in 1T-TaSSe ex-
hibi s i egula a angemen s o SoD clus e s due o he compe i ion be-
ween di e en s acking sequences and he chemical subs i u ion o Se o
S. This leads o sho - ange CDW co ela ions and a lack o global pe i-
odici y, cha ac e is ics ha we deem mo e accu a ely con eyed by he e m
"mosaic" a he han simply "incommensu a e." The use o his e m empha-
sizes he s uc u al agmen a ion and local o de inhe en o he ICCDW
in his ma e ial. This mosaic s uc u e is appa en in he opog aphical im-
age showed in Fig. 4.14
The i s s ep in he expe imen al ou ine ca ied by ou collabo a o s in-
ol es cha ac e izing he la ge-scale elec onic s uc u e o he mosaic phase
in o de o unde s and i s me allic beha iou . This is achie ed h ough
dI/dV spec oscopy and con ex ualized by using DFT calcula ions ha we
p esen in he ollowing. A b ie in oduc ion o he undamen als o his
expe imen al echnique, ollowing Re . [243] is p esen ed in App.C.8. In
sho , we can link dI/dV o he local DOS (LDOS) in a egion o he sam-
ple, his is showcased in Fig. 4.16 whe e we show he ag eemen be ween
DFT calcula ions o he DOS and dI/dV expe imen al measu emen s. A
ep esen a i e igu e on he opog aphy o he mosaic phase is p esen ed
in Fig. 4.14, whe e he domain walls a e ma ked wi h a yellow line and he
mosaic phase is explici .
The esul s om he dI/dV spec oscopy a e p esen ed in in Fig. 4.15.
The o ange dI/dV spec um shown in Fig. 4.15 (a) quali a i ely cap u es
he essen ial ea u es obse ed in he mosaic phase o 1T-TaSSe: he occu-
pied s a e egion ( ep esen ed by he nega i e bias) shows a ea u eless,
inc easing DOS and he emp y s a e egion (posi i e bias) p esen s wo
p onounced wide peaks C1and C2 o ene gies Vb< 0.6V. These peaks a e
consis en ly obse ed ac oss di e en egions o he sample and emain
app oxima ely equidis an wi hin he ange EF< Vb< 0.6V. This beha io
is u he illus a ed by he g ay dI/dV spec um in Fig. 4.15, acqui ed a a
di e en spa ial loca ion.
4.4.3Why s acking diso de ?
We a ibu e he a ia ion o he obse ed emp y-s a e esonances (di e -
ences be ween he o ange and g ay lines in Fig. 4.15) o changes in he
e ical s acking o he CDW be ween laye s, as p e iously epo ed in 1T-
TaS2[241]. We u he hypo hesize ha his andom CDW s acking cons i-
u es he main mechanism behind me alliza ion in 1T-TaSSe by a comp en-
hensi e analysis o pas li e a u e ha we de ail in he ollowing.
This hypo hesis is oo ed in p e ious s udies on isos uc u al compounds.
4.4 expe imen al ealiza ion:me allici y h ough andom s acking.130
a
bc
2.4 Å
0
09 µS
Topog aphy
T = 2 K
Conduc ance, Vs = 0 V (EF)
-0.4 -0.2 0.0 0.2 0.4 0.6 0.8
0
2
dI/dV (A b. Uni s)
Bias ol age (V)
1
-400 mV 5 mV 620 mV
Conduc ance
min
max
230 mV
Figu e 4.14: Expe imen al opog aphy (Vs=0.37 V,I=0.3 nA). Domain walls a e
ma ked wi h a yellow line. Adap ed om Re . [244].
Fo ins ance, in 1T-TaS2, e ical CDW s acking o ms s ong AA dime s
(see Fig. 4.5), which ende he sys em insula ing bo h in pe iodic and non-
pe iodic dime s acking sequences. By con as , 1T-TaSe2adop s a pe iodic
AC s acking sequence, shows no dime iza ion, and exhibi s me allic beha -
io [239]. While in he p e ious Sec ion he choice o chalcogen a ec s he
o e all cha ge ans e , we see ha he e i plays a c ucial ole in de e min-
ing he s acking a angemen . This sugges s ha he s acking sequence in
he andomized-calchogen alloy 1T-TaSSe is inhe en ly complex and un-
likely o ollow a simple pa e n. Ou hypo hesis is u he suppo ed by
he wo k o he g oup om Y. Zhang [216] showing ha s acking andom-
iza ion in he me as able mosaic phase o 1T-TaS2leads o me alliza ion.
Mo eo e , s able pa ches o he mosaic phase ha e been obse ed in equi-
lib ium 1T-TaS2[245], suppo ing ha hey can spon aneously happen in
1T-TaSSe.
In he case o 1T-TaS2−xSex, he andom dis ibu ion o Se a oms ac oss di -
e en laye s likely pins CDW domains independen ly (each domain sepa-
a ed by he yellow line in Fig. 4.14), na u ally leading o s acking diso de .
This mechanism is also ele an in ela ed alloys such as 1T-TaxZ 1−xSe2,
whe e alloy-induced diso de dis up s CDW s acking [246]. In 1T-TaSSe,
his e ec is expec ed o be e en s onge , since he sys em s a is ically in e -
pola es be ween he dime ized 1T-TaS2and he me allic 1T-TaSe2, making
all s ackings nea ly equally p obable. Gi en ha bo h ARPES and anspo
4.4 expe imen al ealiza ion:me allici y h ough andom s acking.131
a
bc
2.4 Å
0
09 µS
Topog aphy
T = 2 K
Conduc ance, Vs = 0 V (EF)
-0.4 -0.2 0.0 0.2 0.4 0.6 0.8
0
2
dI/dV (A b. Uni s)
Bias ol age (V)
1
-400 mV 5 mV 620 mV
Conduc ance
min
max
230 mV
Figu e 4.15: Elec onic s uc u e o he mosaic CDW phase. Uppe panel, conduc-
ance maps aken in he same egion o a CDW domain a di e en
bias ol ages. Lowe panel, dI/dV spec a acqui ed in di e en loca-
ions in CCDW domains o he mosaic phase (Vac =5mV,T=4.2K).
measu emen s show me alliza ion o Se con en x>0.8, and in line wi h
p io heo e ical and expe imen al wo k, we conclude ha me alliza ion in
hese alloys could p ima ly a ise om andom CDW s acking.
4.4.4Why no domain walls?
Fu he suppo o his in e p e a ion comes om uling ou domain walls
hemsel es as he p ima y cause o me allici y, since some s udies poin ed
hem as he eason o supeconduc i i y in hese compounds [230,231].
To in es iga e his, ou collabo a o s pe o med spa ially esol ed dI/dV
mapping o he mosaic phase wi hin a bias ange o ±1eV, as shown
in he uppe panel o Fig. 4.15. These measu emen s e eal ha he con-
duc ance is p edominan ly localized a ound he cen al a om and i s six
nea es neighbo s in each SoD clus e , pa icula ly in he ene gy ange
−0.35 V< Vb< 0.55 V, as illus a ed in he cen al conduc ance maps. This
o bi al ex u e is quali a i ely simila o ha obse ed in 1T-TaS2[216,247]
and highligh s he dominan ole o Ta-de i ed dz2o bi als nea EF. Com-
bined wi h STS da a showing a ini e DOS a EF, his obse a ion con i ms
he me allic cha ac e o he sys em and suppo s ou in e p e a ion.
4.4 expe imen al ealiza ion:me allici y h ough andom s acking.132
a
b
c
-0.4 -0.2 0.0 0.2 0.4 0.6 0.8
0
2
dI/dV (A b. Uni s)
Ene gy (eV)
1
0
2
4
a
b
c
0
2
4
a
b
c
DOS (A b. Uni s) DOS (A b. Uni s)
AB CDW s acking
B
A
BC
A
A
BC
AA CDW s acking
AA CDW s acking
AB CDW s acking
Expe imen
C1C2
C1C2
C1
C2
ΓM K Γ
−1.0
−0.5
0.0
0.5
1.0
E - EF(eV)
AA A TOP
0.00
0.02
0.04
0.06
0.08
0.10
ΓM K Γ
−1.0
−0.5
0.0
0.5
1.0
E - EF(eV)
AB A TOP
0.00
0.02
0.04
0.06
0.08
0.10
d
a
b
c
-0.4 -0.2 0.0 0.2 0.4 0.6 0.8
0
2
dI/dV (A b. Uni s)
Ene gy (eV)
1
0
2
4
a
b
c
0
2
4
a
b
c
DOS (A b. Uni s) DOS (A b. Uni s)
AB CDW s acking
B
A
BC
A
A
BC
AA CDW s acking
AA CDW s acking
AB CDW s acking
Expe imen
C1C2
C1C2
C1
C2
ΓM K Γ
−1.0
−0.5
0.0
0.5
1.0
E - EF(eV)
AA A TOP
0.00
0.02
0.04
0.06
0.08
0.10
ΓM K Γ
−1.0
−0.5
0.0
0.5
1.0
E - EF(eV)
AB A TOP
0.00
0.02
0.04
0.06
0.08
0.10
d
d
a
b
c
-0.4 -0.2 0.0 0.2 0.4 0.6 0.8
0
2
dI/dV (A b. Uni s)
Ene gy (eV)
1
0
2
4
a
b
c
0
2
4
a
b
c
DOS (A b. Uni s) DOS (A b. Uni s)
AB CDW s acking
B
A
BC
A
A
BC
AA CDW s acking
AA CDW s acking
AB CDW s acking
Expe imen
C1C2
C1C2
C1
C2
ΓM K Γ
−1.0
−0.5
0.0
0.5
1.0
E - EF(eV)
AA A TOP
0.00
0.02
0.04
0.06
0.08
0.10
ΓM K Γ
−1.0
−0.5
0.0
0.5
1.0
E - EF(eV)
AB A TOP
0.00
0.02
0.04
0.06
0.08
0.10
d
a
b
c
-0.4 -0.2 0.0 0.2 0.4 0.6 0.8
0
2
dI/dV (A b. Uni s)
Ene gy (eV)
1
0
2
4
a
b
c
0
2
4
a
b
c
DOS (A b. Uni s) DOS (A b. Uni s)
AB CDW s acking
B
A
BC
A
A
BC
AA CDW s acking
AA CDW s acking
AB CDW s acking
Expe imen
C1C2
C1C2
C1
C2
ΓM K Γ
−1.0
−0.5
0.0
0.5
1.0
E - EF(eV)
AA A TOP
0.00
0.02
0.04
0.06
0.08
0.10
ΓM K Γ
−1.0
−0.5
0.0
0.5
1.0
E - EF(eV)
AB A TOP
0.00
0.02
0.04
0.06
0.08
0.10
d
d
Figu e 4.16: (a) and (b), ske ch (le ) and calcula ed band s uc u e o a bilaye
T/T TaSSe in he CDW phase wi h AA and AB s acking espec i ely
( igh ). (c) Co esponding DOS o he calcula ed s uc u es in (a) and
(b) (uppe an middle panel, espec i ely). The lowe panel shows he
dI/dV spec a acqui ed in he mosaic phase. Adap ed om Re . [244]
4.4.5DFT s udy
In o de o in e p e hese expe imen al esul s and alida e ou hypo he-
sis, we pe o med a DFT calcula ion o a T/T bilaye o TaSSe wi h di e en
CDW s ackings (namely, AA and AB11 None heless, hey a e no included
in he main ex because he expe imen al measu emen s om ou collab-
o a o s we e pe o med in domains wi h AA and AB s acking.). The goal
o hese calcula ions is wo old: i s ly, o link he a ia ion o he emp y-
s a e esonances o he spa ially changing e ical s acking o he CDW.
Secondly, o con i m ha he DOS is loca ed a ound he SoD wi hin he
CDW domains.
All DFT calcula ions we e pe o med using VASP [67,68] .6.2.1wi h p ojec o -
augmen ed wa e pseudopo en ials and GGA wi h PBE pa ame iza ion
[74]. Sel -consis en calcula ions we e ound o be well con e ged o a ki-
ne ic cu o o 420eV and a 15 ×15 ×1 k-mesh sampling. Fa -band o bi al-
p ojec ed bands uc u es we e ob ained using PyP oca [223] and 2D pa -
ial cha ge densi y maps we e ob ained by in eg a ing he cha ge densi y e-
11 We did also pe o m calcula ions o he AC s acking ha we p esen in App. C.10.1.
4.4 expe imen al ealiza ion:me allici y h ough andom s acking.133
a
b
c
-0.4 -0.2 0.0 0.2 0.4 0.6 0.8
0
2
dI/dV (A b. Uni s)
Ene gy (eV)
1
0
2
4
a
b
c
0
2
4
a
b
c
DOS (A b. Uni s) DOS (A b. Uni s)
AB CDW s acking
B
A
BC
A
A
BC
AA CDW s acking
AA CDW s acking
AB CDW s acking
Expe imen
C1C2
C1C2
C1
C2
ΓM K Γ
−1.0
−0.5
0.0
0.5
1.0
E - EF(eV)
AA A TOP
0.00
0.02
0.04
0.06
0.08
0.10
ΓM K Γ
−1.0
−0.5
0.0
0.5
1.0
E - EF(eV)
AB A TOP
0.00
0.02
0.04
0.06
0.08
0.10
d
a
b
c
-0.4 -0.2 0.0 0.2 0.4 0.6 0.8
0
2
dI/dV (A b. Uni s)
Ene gy (eV)
1
0
2
4
a
b
c
0
2
4
a
b
c
DOS (A b. Uni s) DOS (A b. Uni s)
AB CDW s acking
B
A
BC
A
A
BC
AA CDW s acking
AA CDW s acking
AB CDW s acking
Expe imen
C1C2
C1C2
C1
C2
ΓM K Γ
−1.0
−0.5
0.0
0.5
1.0
E - EF(eV)
AA A TOP
0.00
0.02
0.04
0.06
0.08
0.10
ΓM K Γ
−1.0
−0.5
0.0
0.5
1.0
E - EF(eV)
AB A TOP
0.00
0.02
0.04
0.06
0.08
0.10
d
d
a cb d
Vs = - 400 mV Vs = +5 mV Vs = +232 mV Vs = +620 mV
2.3 nS0 0.5 nS0 1.3 nS0 2.2 nS0
9 Å
a cb d
Vs = - 400 mV Vs = +5 mV Vs = +232 mV Vs = +620 mV
2.3 nS0 0.5 nS0 1.3 nS0 2.2 nS0
9 Å
a b
C1
C2
Figu e 4.17: (a) DFT spa ial maps o he C1 (le panels) and C2 ( igh panels)
peaks in he AA (uppe panels) and AB (lowe panels) s acking. (b)
Spa ial mapping o he o bi al ex u e o he CDW a selec ed bias,
a ound g ay C1 (uppe panel) and o ange C1 (lowe panel).
sul ing om sel -consis en calcula ions in in e als o 0.05 −0.10eV a ound
he C1and C2peaks o he LDOS in bo h s ackings. The T/T−TaSSe bilaye
s uc u e is buil om 1T −TaSe2CDW √13 ×√13 laye s wi h 50% an-
domized Sa oms in chalcogen posi ions [248]. Sepa a ion be ween laye s
is chosen o be commensu a e wi h bulk and se o 6.29Å o in e -laye
Ta −Ta dis ances [225].
Ou esul s success ully ep oduce he main quali a i e ea u es o he
expe imen al DOS as well as he peak-posi ion dependence on s acking as
shown in Fig. 4.16 (c). Fig. 4.16 (a) and (b) display he A−a om p ojec ed
elec onic band s uc u e o he op 1T laye in bo h AA and AB s ackings
espec i ely. The band s cu es o bo h con igu a ions exhibi simila ea-
u es: wo almos la bands, abo e and below he Fe mi le el, lying in he
gap be ween wo se s o dispe si e en angled bands loca ed a g ea e and
lowe ene gies. These bands can be easy o iden i y when compa ing wi h
he bands uc u es om he monolaye s p esen ed in Fig. 4.3. The a om-
p ojec ed DOS cu es displayed in uppe and middle panels in Fig. 4.16.
These peaks co espond, om le o igh , o he wo la bands and he
i s se o en angled conduc ion bands. In o de o acili a e he compa i-
son wi h expe imen al dI/dV esul s, he peaks associa ed wi h he uppe
la band and he i s g oup o conduc ion bands ha e been labeled as C1
and C2, espec i ely. The compa ison be ween DFT, calcula ed DOS and
4.4 expe imen al ealiza ion:me allici y h ough andom s acking.134
he expe imen al dI/dV measu emen s show subs an ial ag eemen 12, in
pa icula when examining he ene gy gap be ween he peaks C1and C2.
The sepa a ion be ween he unoccupied la band and he nea es conduc-
ion band in DFT is g ea e o AA s acking ∆EAA ≈0.30eV, han o AB
s acking ∆EAB ≈0.25 eV. This end helps us iden i y he o ange and g ay
dI/dV cu es in Fig.4.15 o AA and AB s acking espec i ely. Upon his
iden i ica ion, his end is co obo a ed by dI/dV measu emen s, whe e
∆EAA < 0.30eV and ∆EAB > 0.30 eV. Mo eo e , he 2D pa ial cha ge maps
in Fig. 4.17 (a) u he mo e essemble he conduc ance maps a Fig. 4.17 (b)
u he con i ming he lozalized na u e om C1and C2peaks and show-
casing he ag eemen be ween DFT and expe imen al esul s.
Due o he ag eemen be ween DFT and expe imen al da a, we can suc-
ces ully associa e he a ia ion o he obse ed emp y-s a e esonances (di -
e ences be ween he o ange and g ay lines in Fig. 4.15) o changes in he
e ical s acking o he CDW be ween laye s.
4.4.6Concluding ema ks o his Sec ion
In summa y, h ough a combined expe imen al and heo e ical in es iga-
ion, we ha e demons a ed ha he me allic beha io o 1T-TaSSe a ises
p ima ily om s acking diso de in he CDW phase. STM/STS spec oscopy
e ealed local a ia ions in he elec onic s uc u e, which we we e able o
associa e wi h dis inc s acking con igu a ions h ough i s -p inciples DFT
calcula ions. These esul s ule ou domain walls as he main o igin o
me allici y and ins ead highligh andom CDW s acking ( a o ed by he
s uc u al complexi y o he alloy and he dis ibu ion o chalcogen a oms)
as he key mechanism enabling in e laye unneling and he eme gence o
a ini e DOS a he Fe mi le el a ising om bulk e ec s. The consis ency
be ween he obse ed spec al ea u es and he calcula ed band s uc u es
and pa ial cha ge densi ies ein o ces his in e p e a ion and p o ides a mi-
c oscopic unde s anding o how a nominally insula ing laye ed sys em be-
comes me allic (and, unde a o able condi ions, supe conduc ing) h ough
s acking-d i en me alliza ion.
The de ails o supe conduc i i y cha ac e iza ion along some u he expe -
imen la de ails beyond he scope o his Thesis a e p esen ed in App.C.10.
12 E en when hese calcula ions e lec he main quali a i e ea u es o he su ace DOS o
a gi en s acking o he ou e mos laye s, we could no ind ini e DOS a he Fe mi le el.
This ea u e is only eco e ed in a bulk calcula ion wi h a dis o ed s acking a angemen .
4.5 concluding ema ks 135
4.5 concluding ema ks
In conclusion, his Chap e has p o ided a de ailed explo a ion o he
elec onic and s uc u al p ope ies o TMDs and hei he e os uc u es,
shedding ligh on hei ele ance o co ela ed phenomena such as Mo
and Kondo physics, CDW ansi ions and supe conduc i i y. Al hough his
wo k is p ima ily based on ab ini io single-pa icle app oaches, we ha e es-
ablished a heo e ical amewo k ha is ins umen al in unde s anding he
complex in e ac ions p esen in hese sys ems.
The sys ema ic s udy o cha ge ans e in T/H bilaye s o MX2and he
4Hb−TaX2s uc u e (M=Nb,Ta;X=S,Se) s ands as he main heo e ical
con ibu ion o his Chap e . Ou esul s indica e ha nei he Hubba d U
no an de Waals co ec ions signi ican ly a ec cha ge ans e , and we
ule ou in e laye dis ance as he p ima y con olling pa ame e . Ins ead,
we ha e iden i ied clea ends ac oss di e en compounds: gene ally, Se-
based sys ems exhibi highe cha ge ans e han hei S-based coun e -
pa s, while Ta-based compounds end o show g ea e cha ge ans e
compa ed o hei Nb-based analogs. Mo eo e , we p edic ha cha ge
ans e is mo e p onounced in bulk 4Hbhe e os uc u es han in bilaye s.
These esul s p o ide aluable insigh s in o he b oade phenomenology
o cha ge ans e in T/H sys ems and i s implica ions o elec onic co e-
la ions. F om an expe imen al pe spec i e, ou indings help con ex ualize
p e ious obse a ions o Kondo beha io in T/H bilaye s, pa icula ly in
TaSe2[40,41], TaS2[42,43], and NbSe2[44,45], as well as di e ences in
he o bi al cha ac e o highe -ene gy s a es [40]. Ra he han indica ing
inconsis encies be ween expe imen s, hese a ia ions may e lec in insic
di e ences in cha ge ans e and p oximi y o a po en ial Mo insula ing
s a e, emphasizing he need o p ecise expe imen al e i ica ion o cha ge
ans e ac oss indi idual compounds. Addi ionally, ou s udy sugges s
ha 4Hb-TaS2exhibi s a la ge cha ge ans e , wi h clea e e idence o a
nea ly unoccupied la band [51,52,56], which plays a mino ole in su-
pe conduc i i y. Howe e , ou calcula ions indica e ha in 4Hb-TaSe2, he
la band may be mo e popula ed, po en ially ha ing a s onge in luence
on he supe conduc ing s a e [227–229]. This inding opens new di ec ions
o u u e esea ch on he ole o la -band physics in T/H he e os uc u es
and i s implica ions o uncon en ional supe conduc i i y.
Fu he mo e, he combined e o be ween ou expe imen al collabo a o s
and he DFT calcula ions, in o med by he insigh s gained in Sec. 4.3, has
allowed us o iden i y he o igin o me allici y in he a p io i insula ing
1T-TaSSe. Ou esul s indica e ha he me allic beha io a ises om he
local andomiza ion o CDW s acking, while he p esence o domain walls
wi hin indi idual laye s appea s o ha e li le impac on he eme gence
4.5 concluding ema ks 136
o he me allic s a e. These indings enabled ou collabo a o s o u he
cha ac e ize and in es iga e he supe conduc ing p ope ies o he sys em,
showcasing he s eng h o he in e play be ween heo e ical modeling and
expe imen al obse a ion.
Taken oge he , hese esul s highligh he c i ical ole o cha ge ans e , in-
e laye in e ac ions, and s uc u al diso de in de e mining he elec onic
p ope ies o TMDs and hei he e os uc u es. Fu u e wo k should aim
o sys ema ically map ou cha ge ans e ac oss he en i e amily o T/H
compounds, explo e doping and s ain enginee ing as po en ial uning pa-
ame e s, and u he in es iga e he connec ion be ween s acking diso de
and co ela ed e ec s. Expanding he chemical and s uc u al phase space
o hese ma e ials may lead o he disco e y o no el eme gen quan um
phases, pa ing he way o u u e s udies a he in e sec ion o s ong co -
ela ions, opology, and low-dimensional supe conduc i i y.
