TESE DE DOUTORAMENTO
INTERPLAY BETWEEN
ELECTRONIC, MAGNETIC AND
STRUCTURAL DEGREES OF
FREEDOM IN CRYSTALLINE
TRANSITION METAL
COMPOUNDS
Adol o O e o Fumega
ESCOLA DE DOUTORAMENTO INTERNACIONAL DA UNIVERSIDADE DE SANTIAGO DE
COMPOSTELA
PROGRAMA DE DOUTORAMENTO EN CIENCIA DE MATERIAIS
SANTIAGO DE COMPOSTELA
2021
DECLARACIÓN
DO AUTOR/A DA TESE
In e play be ween Elec onic, Magne ic and S uc u al
Deg ees o F eedom in C ys alline T ansi ion Me al
Compounds
D./Dna.
Adol o O e o Fumega
P esen o a miña ese, seguindo o p ocedemen o axei ado ao Regulamen o, e decla o que:
1) A ese aba ca os esul ados da elabo ación do meu aballo.
2) De selo caso, na ese aise e e encia ás colabo acións que i o es e aballo.
3) A ese é a e sión de ini i a p esen ada pa a a súa de ensa e coincide coa e sión en iada en
o ma o elec ónico.
4) Con i mo que a ese non inco e en ningún ipo de plaxio dou os au o es nin de aballos
p esen ados po min pa a a ob ención dou os í ulos.
En San iago de Compos ela, 13 de Maio de 2021
Asdo.
...........
AUTORIZACIÓN
DOS
DIRECTORES
/
TITOR
DA TESE
In e play be ween Elec onic, Magne ic and
S uc u al Deg ees o F eedom in C ys alline
T ansi ion Me al Compounds
D./Dna.
Víc o Pa do Cas o
D./Dna. José
F ancisco Ri adulla Fe nández
INFORMA/N:
Que a p esen e ese, co espóndese co aballo ealizado po D/Dna. Adol o O e o
Fumega, baixo a nosa di ección, e a
u o izamos
a súa
p esen ación
,
conside ando
que eúne os
equisi os
esixidos no R
egulamen o
de Es udos de
Dou o amen o da USC,
e
que
como di ec o es des a
non inco e nas causas
de
abs ención
es ablecidas
na Lei
40/2015.
De aco do co indicado no Regulamen o de Es udos de Dou o amen o, decla amos
amén que a p esen e ese de dou o amen o é idónea pa a se de endida en base á
modalidade Monog á ica con ep oducción de publicaciones,
nos que a
pa icipación do/a dou o ando/a oi decisi a pa a a súa elabo ación e as
publicacións se axus an ao Plan de In es igación.
En San iago de Compos ela, 13 de Maio de 2021
Abs ac
In e play be ween Elec onic, Magne ic and S uc u al
Deg ees o F eedom in C ys alline T ansi ion Me al
Compounds
Adol o O e o Fumega
i
Adol o O e o Fumega
ii
C ys alline T ansi ion Me al Compounds (TMC) show a di e se amoun o physical
p ope ies. They can be good me als, semiconduc o s o insula o s, and hey can de elop
in iguing o de ed phases such as e omagne ism o supe conduc i i y. The pa ially
illed d elec ons p o ided by he ansi ion me al a om play a undamen al ole o de e -
mine hese. I is he sub le in e play be ween hei elec onic, magne ic and s uc u al
deg ees o eedom ha leads o di e en g ound s a e p ope ies and phase ansi ions.
This he e ogeneous physical beha iou makes TMC highly in e es ing om a echnolog-
ical poin o iew, bu also om a mo e undamen al pe spec i e. The e o e, his kind
o compounds ep esen an ideal pla o m o s udy eme gen phenomena in Condensed
Ma e Physics. In his hesis we ha e s udied om a heo e ical poin o iew he in-
e play be ween he elec onic, magne ic and s uc u al deg ees o eedom in TMC. The
i s h ee chap e s a e de o ed o explain he heo e ical aspec s and me hods equi ed
o pe o m he di e en s udies in he subsequen chap e s. In wha ollows, we will
summa ize he con en and esul s o he chap e s p esen ed in his hesis.
In Chap e 1 we p o ide he heo e ical aspec s o desc ibe ma e ials a he nanoscale.
Nuclei and elec ons a e he undamen als pa icles ha de e mine he p ope ies o ma e-
ials. An unsol able many-body in e ac ing Hamil onian is p esen ed as he co ne s one
o desc ibe a eal ma e ial. The e o e, di e en app oxima ions a e equi ed o make
he p oblem ac able. We s a wi h he Bo n-Oppenheime and adiaba ic app oxima-
ions ha allow o sepa a e he analysis o nuclei and elec ons. The mul ielec onic
p oblems is discussed in he amewo k o he Densi y Func ional Theo y (DFT). This
heo y is de eloped upon he Hohenbe g-Kohn heo ems, whe e he elec onic densi y is
demons a ed o uniquely de e mine he p ope ies o a ma e ial. DFT is made p ac ical
h ough he Kohn-Sham app oach. An independen elec on sys em is used o mimic he
in e ac ing one. Howe e , in his o mula ion he elec on co ela ion, in oduced ia he
so-called exchange-co ela ion e m, emains elusi e. Hence, app oxima ions o his e m
a e equi ed. We p o ide de ini ions o di e en exchange-co ela ion unc ionals. A e
ha , we p esen he me hods used o sol e he Kohn-Sham equa ions, om ull-elec on
localized-o bi al app oaches o pseudopo en ial plane-wa e me hods. Mo eo e , simpli ied
models, like spin o igh -binding Hamil onians, a e in oduced a his s ep. Rega ding
he nuclea mo ion, a classical la ice dynamic app oach is in oduced. We desc ibe i s
he ha monic app oxima ion o ob ain he phonon spec um o a c ys al. A e ha , we
see how o compu e anha monic e ec s. This will be essen ial o co ec ly desc ibe he
la ice hea anspo . Di e en me hods o compu e he la ice dynamics a e exposed.
The e o e, in his i s chap e we pa e he way o s udy om a heo e ical poin o iew
any ma e ial.
In Chap e 2, he undamen al concep o symme y is analyzed. We s a p o iding
a de ini ion o symme y g oup. A e ha we desc ibe he di e en symme y g oups
ha a e encoun e ed in a c ys al, and in pa icula when dealing wi h TMC. A de ailed
exposi ion o di e en poin g oups and hei e ec on delec ons is gi en. T ansla ional
symme y is p esen ed as a undamen al symme y in c ys als. The no ion o (elec onic
and phonon) ene gy bands is shown o be de e mined by his symme y. He e, we p o ide
iii
Adol o O e o Fumega
x
Os compos os c is alinos con me ais de ansición, TMC das súas siglas en inglés,
amosan p opiedades ísicas moi di e sas. Son dende bos me ais a a semicondu o es ou
aillan es, e moi os de eles desen ol en ases o denadas como o e omagne ismo ou a su-
pe condu i idade. Os me ais de ansición achegan elec óns dpa cialmen e ocupados,
se án es es os que xoguen un papel undamen al á ho a de de e mina as p opiedades
des es ma e iais. A su il in e acción en e os g aos de libe dade elec ónicos, es u u ais e
magné icos é a esponsable de es ablece di e en es es ados undamen ais e de ocasiona
ansicións en e ases o denadas. Es e compo amen o an he e oxéneo ai que os TMC
sexan al amen e in e esan es dende un pun o de is a ecnolóxico, pe o amén dende unha
pe spec i a pu amen e eó ica. Es es compos os ep esen an unha excelen e pla a o ma
pa a es uda enómenos eme xen es en Física da Ma e ia Condensada. Nes a ese es uda-
mos dende un pun o de is a eó ico a in e acción en e os g aos de libe dade elec ónicos,
es u u ais e magné icos nos TMC. Os p imei os es capí ulos es án adicados a explica
os undamen os e mé odos eó icos necesa ios pa a le a a cabo os es udos da segunda
pa e. No que segue esumi emos o con ido e os esul ados de cada un dos capí ulos.
No capí ulo 1 amosamos como se desc iben os ma e iais na nanoescala. Elec óns e
núcleos son as pa ículas undamen ais que de e minan as p opiedades dos ma e iais a a-
és das súas in e accións. Es ás eñen desc i as po un Hamil oniano i esoluble de moi os
co pos. Polo an o, se á p eciso ealiza nume osas ap oximación pa a ace o p oblema
a able. Comezamos in oducindo as ap oximacións de Bo n-Oppenheime e a adiabá i-
ca. Es as pe mi en sepa a a análise pa a elec óns e núcleos. O p oblema mul ielec ónico
discu i émolo no ma co da eo ía do uncional da densidade, DFT do inglés. Es á eo ía
baséase nos eo emas de Hohenbe g e Kohn que demos an que a densidade elec ónica
de e mina de xei o uní oco as p opiedades do ma e ial. O xei o de ace p ác ica es a
eo ía é po medio da ap oximación de Kohn e Sham, que conside a que o p oblema de
elec óns in e accionan es se pode mapea nun de elec óns independen es. Non obs an e,
nes a o mulación a co elación en e os elec óns, incluída a a és do e mo de co ela-
ción e in e cambio, é descoñecida e polo an o se án p ecisas ap oximacións a es e e mo.
Amosa emos di e en es ap oximacións ao uncional de co elación e in e cambio. Despois
p esen a emos os mé odos emp egados pa a esol e as ecuación de Kohn-Sham, dende
ap oximacións con o bi ais localizados e ódolos elec óns a a ondas planas e pseudo-
po enciais. Ademáis, in oduci emos nes e pun o un pa de modelos simpli icados: Ha-
mil onianos de espín e modelos ipo igh -binding. Con espec o á dinámica nuclea es a
se á ap oximada dun xei o clásico. Desc ibi emos p imei o a ap oximación ha mónica que
pe mi e ob e o espec o de onóns do c is al. Despois e emos como in oduci e ec os
anha mónicos. Es es se án esenciais pa a desc ibi axei adamen e o anspo e de calo
po onóns. Amosa emos di e en es mé odos pa a calcula a dinámica nuclea . Polo an o,
es e p imei o capí ulo sen a á as bases eó icas no es udo de ma e iais.
No capí ulo 2 analizamos o concep o undamen al de sime ía. Comeza emos dando
unha de inición de g upo de sime ía pa a despois desc ibi os di e en es g upos de sime-
ía que se a opan nun c is al e, máis en pa icula , nos TMC. P opo ciona emos unha
exposición de allada do e ec o causado nos elec óns dpo di e en es g upos pun uais de
xi
Adol o O e o Fumega
sime ía. P esen a emos logo o sime ía aslacional como unha das sime ías máis unda-
men ais dun c is al. Ve emos que a noción de bandas de ene xía (elec ónicas e onónicas)
eme xe des a sime ía. Amosa emos a is a de páxa o di e en es ca ac e ís icas que po-
demos a opa nas bandas dun c is al. Seguidamen e, de inimos a sime ía de in e sión
empo al conec ándoa con di e en es o des magné icos. Po úl imo, p esen a emos unha
b e e in odución ao eno ado ema de in a ian es opolóxicas en ma e ia condensada.
No capí ulo 3 es ablece emos a conexión en e a nanoescala e a mac o escala dun
ma e ial po medio da mecánica es a ís ica e as p opiedades de anspo e. Comeza emos
de inindo can idades especias no equilib io é mico como a densidade de es ados ou a den-
sidade de es ados p oxec ada. As dis ibucións de Bose-Eins ein e Fe mi-Di ac pa a onóns
i elec óns espec i amen e se án de inidas. Es as dan o núme o de es ados ocupados po
unidade de olume a unha ene xía dada pa a un sis ema en equilib io. A es abilidade dun
sis ema se á discu ida en e mos da densidade de es ados e as uncións de dis ibución. De
seguido le a emos o sis ema ó a do equilib io pa a es uda as súas p opiedades de ans-
po e. Es o a émolo no ma co da ecuación de anspo e de Bol zmann, onde campos
ex e nos modi ica án as uncións de dis ibución a ando de es ablece o equilib io nun
empo de elaxación. Es e empo elacionase cos di e en es p ocesos de collisións en e as
pa ículas que a an de es au a o equilib io. Pa a anspo e elec ónico ap oxima e-
mos o empo de elaxación a unha cons an e, men es que no caso de onóns o empo de
elaxación se á calculado po medio de e mos anha mónicos. Todo is o pe mi i anos de i-
ni os seguin es coe icien es de anspo e: condu i idade eléc ica, condu i idade é mica
elec ónica, coe icien e Seebeck e condu i idade é mica onónica. Tamén discu i emos a
in e acción que exis e en e es es coe icien es pa a de e mina como de bo e moeléc ico
é un ma e ial.
Despois des es es capí ulos nos que expoñemos a eo ía e os mé odos pa a esol e
o p oblema de moi os co pos xa es amos en disposición de comeza os nosos es udos
en TMC. En pa icula , cen a émonos en óxidos complexos e ma e ias lamina es de
an de Waals. Es as clases de compos os p esen a unha g an a iedade de p opiedades
ísicas e poden se sin e izados en mul i ude de es u u as de o ma sinxela. No caso dos
óxidos complexos é salien able a súa g an es abilidade en condicións no mais de p esión
e empe a u a. No ocan e aos ma e iais lamina es de an de Waals, o ei o de es a
o mados po capas pe mi e ex olialos a a chega ao limi e en dúas dimensións. Polo
an o, es as amilias de compos os p esen anse como p ome edo es á ho a de es uda
no os enómenos eme xen es a empe a u a ambien e. Apa e de es es ma e iais, ou o
ipo de compos os se án analizados pa a mos a o impo an e ol dos elec óns d.
Comeza emos no capí ulo 4 es udando as p opiedades de anspo e no S TiO3. Es e
compos o ep esen a un p o o ipo pa a o es udo de di e en es enómenos ísicos. Dexei-
o ema cable, o S TiO3dopado a ópase den o dos mello es ma e iais e moeléc icos.
Es e ipo de ma e iais son capaces de con e e de o ma e icien e un g adien e é mico
nunha di e enza de po encial e ice e sa, polo que son moi in e esan es dende o pun o de
is a da inno ación. Nes e ipo de ma e iais é impo an e que o coe icien e Seebeck sexa
maximizado coa inalidade de op imiza a di e enza de po encial xe ada po un g adien e
xii
é mico dado. A maio es, amén é impo an e que a condu i idade é mica onónica sexa a
mínima posible pa a que así se man eña o g adien e de empe a u a. Como ecollemos no
capí ulo 3, es as p opiedades de anspo e eñen de e minadas pola es u u a elec ónica
e c is alina. Polo an o, o obxec i o des e capí ulo se á es uda dende unha pe spec i a ab
ini io cales son as ca ac e ís icas undamen ais des es coe icien es de anspo e e como
poden se al e adas po medio de dis o sións es u u ais. A p imei a pa e des e es udo
es a á adicada ao coe icien e Seebeck. Analiza emos o e ec o da dexene ación o bi al no
coe icien e Seebeck como unción de dis o sión (comp esión e ensión) da celda unidade
e do ni el de dopaxe. Pa a ace is o cons ui emos un modelo simpli icado pa a o coe i-
cien e Seebeck undamen ado nas ecuacións de Mo e Heikes. In oduci emos un ac o
dependen e da ensión que modeliza á o e ec o causado pola dexene ación dos o bi ais 2g
no coe icien e Seebeck. Pode emos calcula dende DFT es e ac o de dexene ación, des-
acoplando des e modo o e ec o da ensión e a dopaxe debidas á en opía da dexene ación
elec ónica ou debidas a modi icacións da es u u a de bandas. A opa emos que as masas
elec ónicas e ec i as e a dexene ación elec ónica son os ac o es p incipais que de e minan
o coe icien e Seebeck como unción da dis o sión. Aplicando o modelo ao caso do S TiO3
a opamos dous éximes de dopaxe. A baixas dopaxes calque ipo de dis o sión diminúe
o alo do coe icien e Seebeck debido a que o e mo dominan e é o das dexene acións,
polo que calque a dis o sión educe o ac o de dexene ación. A al as dopaxes, o éxime
in e esan e en aplicacións, o e mo ene xé ico domina e a opamos que aplica unha en-
sión aumen a o coe icien e Seebeck. Is o é debido a que a ensión p oduce un aumen o do
olume da celda unidade que modi ica de xei o a o able o ancho de banda e polo an o a
masa e ec i a. Na segunda pa e des e capí ulo analiza emos a condu i idade e mica o-
nónica do S TiO3. Fa emos un esc u inio das p incipais causas do anspo e é mico nas
dúas ases es u u ais do S TiO3e como modi icalas aplicado dis o sións na es u u a. A
al as empe a u as e emos unha ase cúbica e a baixas unha e agonal con o acións dos
oc aed os. Na ase cúbica, de e minamos que as bandas acús icas e o modo pola iple-
men e dexene ado son os p incipais esponsables do anspo e de calo . Unha edución do
olume da celda unidade aumen a a dispe sión dos onóns acús icos e a ene xía dos modos
pola es. Des e xei o inc emen ase a elocidade de g upo e edúcense as colisións onónicas
aumen ando a condu i idade é mica. O compo amen o opos o oco e se aumen amos
o olume da celda unidade. No caso de aplica unha dis osión e agonal a dexene ación
dos modos pola es ómpese baixando en ene xía o modo na di ección ce modi icando a
dispe sión das bandas acús icas. Todo is o causa unha diminución do anspo e é mico.
Na ase a baixa empe a u a a apa ición de o acións oc aéd icas p opo ciona no os g aos
de libe dade a conside a cando analizamos a condu i idade é mica. Do mesmo xei o
que oco e na ase cúbica, os onóns acús icos e pola es son os esponsables do anspo e.
Analizando de xei o independen e o e ec o do olume da celda unidade, o a io c/a e
o ángulo das o acións oc aéd icas obse amos que: i) un inc emen o do olume causa
unha diminución da dispe sión das bandas acús icas e baixa a ene xía dos modos pola-
es, diminuindo a condu i iade é mica; ii) inc emen a o a io c/a educe a dispe sión
das bandas acús icas e con elo a condu i idade é mica; iii) a opamos que o ángulo das
xiii
Adol o O e o Fumega
o ación oc aéd icas es á o emen e acoplado cos modos pola es, un aumen o do ángulo
aumen a a ene xía dos modos pola es diminuíndo as in e accións onónicas e aumen an-
do a condu i idade é mica. Se conside amos conxun amen e as dúas análises pa a os
coe icien es de anspo e, a espos a e moeléc ica pode op imiza se inc emen ando o
olume da celda unidade con ensión e educindo as o acións oc aéd icas. Os esul ados
des e capí ulo p opo cionan un coñecemen o undamen al das p opiedades de anspo e
de óxidos simila es e guían a p ocu a de mello es ma e iais e moeléc icos.
No capí ulo 5 analize emos a ase e omagné ica e aillan e que eme xe no LaCoO3
cando se med a sob e S TiO3na di ección (001). Os cálculos ab ini io que ealiza emos
pe imi án es uda a es abilidade e modinámica de es u u as con dis in as es equiome-
ías nas que in oduci emos acan es de oxíxeno de aco do coas obse acións expe imen-
ais. Se emos capaces de de e mina o es ado undamen al e a es u u a elec ónica e p o-
piedades magné icas do LaCoO3med ado sob e S TiO3. A opamos que a es equiome ía
máis es able é LaCoO2,83; é dici , un 6% de acan es de oxíxeno. Así mesmo, de e mina-
mos que as acan es o man cadeas pe pendicula es á di ección (001); e que o pa áme o
de ede c e as dis ancias en e as capas de á omos de La es án en consonancia coas medi-
das expe imen ais. Dende o pun o de is a elec ónico, e emos que os á omos de Co que
se encon an no plano das acan es son unha mis u a de Co2+ con al o espín e Co2+ con
baixo espín con acoplamen os e omagné icos. Os es an es cobal os man eñense como
Co3+ non magné icos no en o no oc aéd ico. A con igu ación que a opamos da luga a un
momen o magné ico que conco da co expe imen al, así como a un o de e omagné ico.
Con espec o ao compo amen o aillan e, obse amos que un gap ene xé ico eme xe como
consecuencia do desdob amen o do campo c is alino inducido polas ancan e de oxíxeno
a un alo azoable de U. Polo an o, nes e aballo explicamos a ase e omagné ica e
aillan e dende unha pe spec i a iónica moi sinxela. Es es esul ados o ecen unha maio
comp ensión da na u eza dos aillan es e omagné icos e guían a p ocu a de no os óxidos
con es a inusual ase nos que as ancan es de oxíxeno puidesen xoga un papel ele an e.
No capí ulo 6 analiza emos o e ec o da dopaxe con á omos de S na es u u a elec ó-
nica e p opiedades magné icas dos niquela os de capa in ini a RNiO2(R= La, Nd). Es a
amilia de compos os le aba anos p opos a de xei o eó ico como análogo na u al dos
cup a os supe condu o es a al a empe a u a. Non obs an e, a con i mación expe imen al
des e ei o non oco eu a a ai un pa de anos. Os chamados compos os pais; é dici ,
sen dopa , dos cup a os e niquela os amosan un compo amen o elec ónico e magné ico
adicalmen e di e en e. Men es que os cup a os son aillan es an i e omagné icos, os ni-
quela os son me ais e non magné icos. Sen emba go, ao in oduci dopaxe de ocos (dopaxe
con á omos de S ) a supe condu i idade eme xe nas dúas amilias. Polo que a na u eza dos
ocos p esén ase como un es udo a ealiza . Nes e aballo ealiza emos cálculos DFT en
RNiO2como unción do ca ión R (R=La, Nd) e da dopaxe con S (11% e 25%). Emp e-
ga emos dous esquemas di e en es de con axe dob e nos cálculos LDA+U. Os esul ados
amosa án cambios moi subs anciais na es u u a elec ónica des es niquela os ao se en
dopados con S . Po unha pa e, o S educe o e ec o da chamada au odopaxe desp azan-
do as bandas dda e a a a po iba do ni el de Fe mi. Es o da luga a unha desc ipción
xi
cunha sóa banda onde o o bi al dx2−y2do á omo de Ni é dominan e. Ademáis, edúcese
a ene xía ans e encia de ca ga en e os á omos de Ni e O. Con espec o ás p opiedades
magné icas, obse amos que o es ado de baixo espín pa a o Ni2+ é a o able. Polo an o,
odos es es esul ados demos ans que dopando con á omos de S os niquela os adqui en
un compo amen o semellan e ao dos cup a os, suxe indo que a supe condu i idade dos
niquela os p esen a á unha desc ipción como a dos cup a os.
No capí ulo 7 es uda emos a es u u a elec ónica do Cs2CuCl4, un compos o que se
en discu ido no ma co dos líquidos de espín xeome icamen e us ados. Emp ega emos
de xei o combinado écnicas expe imen ais, como a dispe sión inelás ica po esonancia
de aios X (RIXS do inglés) e a espec oscopía po abso ción de aios X (XAS do inglés),
e cálculos pa a de e mina o espec o des e comopos o. A análise mos a que a es u u a
elec ónica es á de eminada na súa maio ía polo desdob amen o dos elec óns ddo Cu2+
a causa do campo c is alino e aéd ico o emen e dis o sionado xe ado polos ións de
Cl1−. Os cálculos DFT pe mi en de e mina os di e en es desdob amen os ene xé icos
obse ados po RIXS. A maio es amén iden i icamos o gap de ans e encia de ca ga
en e o clus e de [CuCl4]2−e os es ados sdo Cs1+. Es e aballo amosa o po encial de
combina expe imen o e eo ía e os esul ados sen an as bases pa a un esc u inio máis
de allado das exó icas p opiedades des e compos o a baixas empe a u as.
Nos seguin es capí ulos achega émonos ao es udo de ma e iais lamina es de an de
Waals. Es e ipo de compos os ep esen a unha pla a o ma excelen e pa a es uda p o-
piedades en ma e ials con baixa dimensionalidade. En pa icula , o magne ismo nes e
lími e de baixa dimensionalidade esul a moi in e esan e polas p ome edo as aplicacións
que pode ía e . No capí ulo 8 p esen a emos un es udo compa a i o en e a es u u a
elec ónica e as p opiedades magné icas de dous compos os de an de Waals e omagné-
icos. Es uda emos o e ec o da dimensionalidade en C B 3e C 2Ge2Te6dende un pun o de
is a expe imen al e eó ico. Pa a es o, no can o de i ao lími e pu amen e bidimensional
es uda emos a e olución aplicando p esión; is o é, índonos a un lími e máis idimensional.
A opa emos que o compos o de C B 3amosa un compo amen o de ipo Mo cun gap
d−de se man én aillan e ao aplica p esión. De modo di e en e, o C 2Ge2Te6p esen a
un gap ene xé ico ipo p−de polo an o unha desc ipción de aillan e po ans e encia
de ca ga. Os nosos cálculos p edín unha ansición aillan e-me al a edo de 6 GPa, que
pode ía es a elacionada coa ansición s uc u al obse ada nos expe imen os de aios
X a al a p esión. O di e en e ca ác e da es u u a elec ónica e conseguin e e olución coa
p esión des es compos os implica unha e olución di e en e das súas p opiedades magné-
icas coa p esión. Os nosos expe imen os amosas unha edución da empe a u a de Cu ie
nos dous compos os ao aplica p esión, non obs an e a súa endencia é di e en e. A nosa
análise eó ica complúe que a di e en e e olución dos acoplamen os magné icos den o
e ó a do plano pa a cada compos o de e mina a di e en e endencia das empe a u as
de Cu ie. Polo que es e aballo esal a o impo an e ol da es u u a elec ónica pa a
explica as in e accións en e espíns que causan a o de e omagné ica. En pa icula , sa-
lien amos o e ec o da p esión, e po an o da dimensionalidade, na es u u a elec ónica
e nas p opiedades magné icas en ma e ias de an de Waals baseados en C .
x
Adol o O e o Fumega
No capí ulo 9 segui emos a es uda ma e iais de an de Waals que se p esen an co-
mo p ome edo es candida os pa a a opa o de e omagné ico de la go alcance no lími e
bidimensional. En conc e o cen a émonos na análise de dous dicalcoxenu os d me ais de
ansición (TMDs do inglés), C Te2e VSe2. Moi os des es compos os desen ol en unha
ase de densidade de ca ga ondula o ia, ou ase cha ge densi y wa e (CDW) en inglés, a
baixas empe a u as que in e acciona co p ocu ado o de e omagné ico. Polo an o o ob-
xec i o p incipal des e capí ulo se á es uda a in e acción en e os o des e omagne icos
e CDW nos TMDs. Na p imei a pa e cen a émonos no C Te2con in ini as capas, un
e omagné ico a empe a u a ambien e cos momen os apun ando den o dos planos de
an de Waals. Despois, pasamos ao es udo do lími e monocapa, é dici bidimensional,
e a opamos que unha ase CDW eme xe a baixas empe a u as. P opo cionamos unha
análise de alla da no a sime ía c is alina e como ecoñecela expe imen almen e. O onón
esponsable des a ansición é iden i icado o que ab e a posibilidade pa a sal a dunha
ase á ou a ac i ando o de andi o modo. Na ase CDW áb ense pseudogaps ene xé i-
cos a edo do ni el de Fe mi e os o bi ais d localízanse máis. Is o ai que os momen os
magné icos apun en ó a do plano, supe ando as es icións impos as polo eo ema de
Me min e Wagne e dando luga a o de e omagné ico de la go alcance na monocapa.
Tamén a opamos que unha dis o sión ensionan e da monocapa causa un e ec o simila ,
localizando os elec óns d e le ando os momen os a apun a ó a do plano. Na segunda
pa e des e capí ulo cen ámonos no es udo do VSe2. Expe imen almen e demós ase que
a o ma mul icapa des e compos o desen ol e unha ase CDW po debaixo de 110 K e
non se epo a ningún ipo de compo amen o magné ico. O noso es udo ab ini io amo-
sa á que a ase CDW des úe o magne ismo no VSe2mul icapa. Es a minuciosa análise
pe mi e econcilia o desaco do en e os expe imen os e cálculos DFT an e io es nos que
o e ec o da CDW non se es aba a conside a ; e que polo an o p edicían de xei o e óneo
un compo amen o e omagné ico nes e ma e ial. A maio es, a es u u a CDW que ob-
i emos é capaz de explica o sal o a opado expe imen almen e no coe icien e Seebeck á
empe a u a de ansición. O es udo da monocapa con inúa a se un campo de in es iga-
ción moi ac i o; sen emba go, nes e aballo amosamos que a p esenza dunha ase CDW
a ec a de xei o nega i o ao e omagne ismo. Polo an o, nes e capí ulo salien amos o pa-
pel undamen al que xoga a es u u a CDW pa a de e mina as p opiedades magné icas
nos TMDs.
Po úl imo, no capí ulo 10 p esen amos un es udo eó ico no BaSn2. Es e compos o
lamina é un semime al cunha liña nodal opolóxica cando se desp eza o acoplamen o es-
pín o bi al. Os ma e iais opolóxicos es án a a ae moi o in e ese debido a que p esen an
es ados supe iciais obus os que pode ían da luga a enómenos eme xen es e a p ome-
edo as aplicacións. Nes e es udo emp ega emos ao BaSn2como p o o ipo pa a in es iga
ansicións opolóxicas inducidas pola p esión. Pa a elo, u iliza emos unha combinación
de cálculos DFT e un modelo igh -binding. A opamos que a p esión uniaxial p oduce
unha ansi ión de ase en e opoloxías non- i iais a edo de 4 GPa. Na ase a al a
p esión o núme o de liñas nodais opolóxicas aumen a. Nes a ansición os elec óns d
xogan un papel undamen al, xa que é o aumen o coa p esión das ene xías de hopping
x i
en e os o bi ais dxz edyz do Ba cos pzdo Sn o que ai eme xe as no as liñas nodais. Ade-
máis, amosamos a o ma de de ec a expe imen almen e es e ipo de ansicións median e
medidas de magne o anspo e.
Os aspec os e mé odos eó icos p esen ados nes a ese, xun o cos es udos nos dis in os
TMC, p opo cionan unha mos a ep esen a i a do coñecemen o undamen al que se es á
a le a a cabo na á ea de Física da Ma e ia Condensada.
x ii
Adol o O e o Fumega
x iii
Con en s
In oduc ion and Objec i es xxiii
I Theo y and Me hods 1
1 The Physical Sys em 3
1.1 A Many-Body Quan um Sys em . . . . . . . . . . . . . . . . . . . . . . . . 3
1.2 Nuclei a e hea ie han Elec ons . . . . . . . . . . . . . . . . . . . . . . . 6
1.3 In e ac ingElec ons.............................. 9
1.3.1 Densi y Func ional Theo y . . . . . . . . . . . . . . . . . . . . . . . 10
1.3.2 Wha abou Elec on Independence? . . . . . . . . . . . . . . . . . 13
1.3.3 Spin and Rela i is ic E ec s . . . . . . . . . . . . . . . . . . . . . . 18
1.3.4 Elec onic S uc u e Me hods . . . . . . . . . . . . . . . . . . . . . 19
1.3.5 Simpli iedModels............................ 24
1.4 NucleiandVib a ions ............................. 26
1.4.1 Ha monicApp oach .......................... 27
1.4.2 Anha monic E ec s . . . . . . . . . . . . . . . . . . . . . . . . . . . 29
1.4.3 Nuclea Dynamics Me hods . . . . . . . . . . . . . . . . . . . . . . 30
2 Symme y 35
2.1 De ini ion o a Symme y G oup . . . . . . . . . . . . . . . . . . . . . . . . 35
2.2 Poin G oupSymme y............................. 36
2.3 T ansla ional Symme y . . . . . . . . . . . . . . . . . . . . . . . . . . . . 39
2.4 Time-Re e sal Symme y . . . . . . . . . . . . . . . . . . . . . . . . . . . . 44
2.5 Topological In a ian s . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 46
3 S a is ics and T anspo 49
3.1 The malEquilib ium.............................. 49
3.2 T anspo P ope ies .............................. 52
xix
Adol o O e o Fumega
in di e en s uc u es. Impo an ly, his kind o compounds end o be ex emely s able
in no mal condi ions due o he oxidized a mosphe e whe e we li e. In ac , many o
he mos in iguing oxides occu na u ally. On he o he hand, laye ed an de Waals
ma e ials a e o med by laye s ha a e bonded ia an de Waals in e ac ions. This weak
bonding allows o g ow and manipula e his kind o compounds in he ew-laye limi .
E en mo e, laye s can be ex olia ed o ob ain a monolaye . This dimensionali y con ol
p omp s o eme gen phenomena. Conside ing all ha , he main body o his hesis will
be de o ed o he s udy o his kind o ma e ials. Howe e , o he classes o compounds
will also be in i ed o exempli y he impo an ole ha d elec ons play in na u e.
Pa II will s a wi h s udies on oxides. Chap e 4 will be de o ed o he analysis o
he he moelec ic p ope ies o S TiO3, in pa icula we will see how he he mopowe
and he la ice he mal conduc i i y can be uned by s uc u al modi ica ions. In Chap-
e 5 we will see how LaCoO3becomes an insula ing e omagne when g own on op o
S TiO3. The e ec o doping in in ini e-laye nickela es will be analyzed in Chap e 6. Re-
cen ly, his class o compounds has been ound o be high- empe a u e-supe conduc ing-
cup a e analogs. In Chap e 7 we make a b eak be o e en e ing in he s udy o laye ed
an de Waals ma e ials. We show how he expe imen al spec um o Cs2CuCl4, a com-
pound discussed in he amewo k o spin liquids, can be explained in combina ion wi h
he me hodology p esen ed in his hesis. A compa a i e s udy be ween he elec onic
and magne ic p ope ies o wo simila C -based an de Waals e omagne s, C B 3and
C 2Ge2Te6, is p o ided in Chap e 8. In Chap e 9 he in e play be ween e omagne ic
and cha ge densi y wa e phases is discussed. Two di e en ansi ion me al dichalco-
genides a e s udied, C Te2and VSe2. Finally, in Chap e 10 we analyzed how d elec ons
can play a ole in opological phase ansi ions by s udying BaSn2unde p essu e. The
a icles whe e hese wo ks ha e been published a e e e ed in he Lis o Publica ions
sec ion, a he end o he hesis.
All he heo e ical aspec s and me hods p esen ed in Pa I, as well as he wo ks
desc ibed in Pa II, will p o ide many o he basics o unde s and he s a e-o - he-a
esea ch in Solid S a e Physics. This hesis will del e in o he undamen als o Ma e ials
Science, and hope ully will se e o achie e some o he global goals ha we ha e discussed
in his in oduc ion.
xx i
Pa I
Theo y and Me hods
1
1 The Physical Sys em
“The mo e we simpli y ou ma e ial
needs he mo e we a e ee o hink o
o he hings."
—Eleano Roose el
In his chap e we will desc ibe om a undamen al poin o iew he physical sys em
ha desc ibes a ma e ial. We will show ha di e en app oxima ions a e needed o deal
wi h ha sys em. Mo eo e , we will p esen he me hods ha we will use in he analysis
o he second pa .
1.1 A Many-Body Quan um Sys em
In he In oduc ion we ha e been alking abou he p ominen ole ma e ials ha e
o ou mode n socie ies. Howe e , we ha e no p o ided ye a o mal de ini ion o wha
a ma e ial is. In e ms o i s cons i uen s a ma e ial is o med by a oms, which we e
i s pos ula ed by Democ i us in Ancien G eece and inally disco e ed by he a he s
o mode n Chemis y An oine La oisie and John Dal on. Thus, he building blocks o
any ma e ial a e he a oms o he di e en elemen s collec ed in he Pe iodic able. The
way in which hese elemen s a e g ouped leads o a ple ho a o ma e ials wi h comple ely
di e en p ope ies ha can be used o echnological applica ions.
I was a he beginning o he 20 h cen u y when i was disco e ed ha a oms a e
o med by nuclei and elec ons. This de ines a leng h scale o nanome e s o desc ibe he
s uc u e o ma e ials, he nanoscale. Elec ons ha e nega i e elec ic cha ge and su ound
he nucleus, and his, in u n, is o med by neu al neu ons and posi i e p o ons. The
numbe o p o ons in he nucleus, called he a omic numbe , de e mines he pa icula
elemen o an a om. Many models we e p oposed o explain he a omic s uc u e, i.e.,
how elec ons and nuclei a e a anged oge he . The Quan um Theo y was de eloped
wi h ha pu pose since elec ons a e a kind o objec ha canno be in e p e ed in e ms
o he Classical heo y. I was Sch ödinge ’s equa ion ha i s ea ed he elec ons in
a sa is ac o y way in e ms o a wa e unc ion [7]. Then, Di ac p o ided a mo e na u al
desc ip ion o spin by w i ing a ela i is ic equa ion [8]. Thus, ma e ials a he nanoscale
3
Adol o O e o Fumega
mus be desc ibed quan um mechanically.
We ha e es ablished ha nuclei and elec ons a e he undamen al pa icles ha
cons i u e ma e ials.1In a quan um mechanical sys em like his, physical s a es o n
elec ons and Nnuclei a e codi ied by a ec o o a complex Hilbe space, usually called
wa e unc ion Ψ( 1, 2, ..., n,R1,R2, ..., RN). This wa e unc ion depends on he coo di-
na es iand RI(including posi ion and spin) o each elec on and nucleus espec i ely,
and can be exp essed as a sum o a comple e se o ss a es Ψs( 1, 2, ..., n,R1,R2, ..., RN)
weigh ed by a p obabili y in ha Hilbe space. The measu able quan i ies will be gi en
by He mi ian ope a o s. The esul s o a measu emen and hei p obabili y o occu can
be compu ed om he wa e unc ion and he eigen ec o s o such ope a o . An essen ial
case o conside is he Hamil onian ˆ
H, which co esponds o he o al ene gy ope a o
o a sys em. The eigen alues and eigen ec o s o he Hamil onian p o ide he possible
alues o he ene gy Esand he s a es |Ψsi ha he sys em can ake:
ˆ
H|Ψsi=Es|Ψsi(1.1)
The Hamil onian is gi en by he sum o he kine ic and po en ial ene gies o all he
pa icles. The e o e, in gene al, we can w i e he Hamil onian o any ma e ial wi h n
elec ons and Nnuclei as
ˆ
H=−
N
X
I=1
~2
2MI∇2
I+1
2
N
X
I=1
N
X
I6=J
ZIZJe2
|RI−RJ|
−~2
2me
n
X
i=1 ∇2
i+1
2
n
X
i=1
n
X
i6=j
e2
| i− j|−
n
X
i=1
N
X
I=1
ZIe2
| i−RI|,
(1.2)
whe e we encoun e 5 e ms well di e en ia ed. The i s is he kine ic ene gy o he
nuclei o mass MI, he second is he Coulomb in e ac ion be ween he di e en nuclei o
cha ge ZI, he hi d one is he kine ic ene gy o elec ons wi h mass me, he ou h e m is
he Coulomb in e ac ion be ween elec ons and he inal e m is he Coulomb in e ac ion
be ween elec ons and nuclei.
No e ha spin e ms ha e been a oided in he many-body Hamil onian w i en in
eq. (1.2). These e ms will be o capi al impo ance o he co ec desc ip ion o many
sys ems such as magne ic and opological ones. We will come back o hem in he ollow-
ing sec ions. The impo an idea ha we wan o highligh a his poin is ha ma e ials
a e desc ibed by many-body Hamil onians, i.e., Hamil onians ha con ain in e ac ions
be ween many di e en pa icles (elec ons and nuclei). Since he wa e unc ion Ψsde-
pends on he se o coo dina es { }and {R}o nelec ons and Nnuclei espec i ely,
we ha e an eno mous amoun o deg ees o eedom coupled by he di e en e ms in
1No e ha he wo d “ undamen al" is usually e e ed o he elemen a y pa icles desc ibed by he
S anda d model in High Ene gy Physics. Howe e , he ene gy scale needed o desc ibe ma e ials is much
lowe , and hence i is ine o conside nuclei and elec ons as he undamen al pa icles o ou heo y.
4
1 The Physical Sys em
Figu e 1.1: Scheme wi h he heo e ical app oach o he many-body p oblem. The di e -
en pa s will be explained in he main ex .
he many body-Hamil onian o eq. (1.2). This ep esen s a highly di icul p oblem o
sol e as soon as we ha e a sys em wi h mo e han wo pa icles (like he hyd ogen a om).
A eal ma e ial is a mac oscopic sys em o med by a numbe o a oms o he o de o
A ogad o’s numbe (NA ≃1023). This makes he p oblem impossible o sol e. Hence,
5
Adol o O e o Fumega
we will need o ind ways o simpli y he many-body Hamil onian. This will be he ask
o he ollowing sec ions. In o de o ha e a a glance all he app oxima ions ha we a e
going o pe o med, we ha e summa ized hem in he ske ch o Fig. 1.1.
1.2 Nuclei a e hea ie han Elec ons
The i s app oxima ion ha we will pe o m on he many-body Hamil onian o eq.
(1.2) is he so-called Bo n-Oppenheime app oxima ion [9]. Taking a close look o ha
Hamil onian, we can obse e ha he nuclea kine ic ene gy e ms a e e y small com-
pa ed o he o he s, as long as he masses MIo he nuclei a e much hea ie han he
elec on mass me. This allows o ea he nuclea kine ic ene gy e ms as a pe u ba ion
upon he Hamil onian o eq. (1.2). The esul leads o he decoupling o he elec onic
({ })and nuclea ({R})deg ees o eedom in he many-body wa e unc ion
Ψs({ },{R}) = X
s0
χss0({R})ψs0({ };{R}),(1.3)
whe e we ha e exp essed he coupled s a es in e ms o a comple e se o s0elec onic
s a es ψs0({ };{R}) o each nuclea con igu a ion in he se {R}. The χss0({R})a e unc-
ions o he nuclea coo dina es and he coe icien s o he elec onic s a es ψs0({ };{R}).
The e o e, we can educe he many-body (nuclea and elec onic) Hamil onian o eq. (1.2)
o a pu ely elec onic Hamil onian ˆ
He. We ha e o neglec he kine ic ene gy o he nu-
clei and conside he nuclea coo dina es as pa ame e s ins ead o a iables. Then, he
Hamil onian educes o
ˆ
He=ˆ
Te+ˆ
Vex +ˆ
Vin +ENuc (1.4)
whe e ˆ
Teis he kine ic ene gy ope a o o elec ons
ˆ
Te=−~2
2me
n
X
i=1 ∇2
i,(1.5)
ˆ
Vex is he po en ial ha ac s on elec ons due o he nuclei, in which posi ions a e
ea ed now as pa ame e s
ˆ
Vex =−
n
X
i=1
N
X
I=1
ZIe2
| i−RI|,(1.6)
ˆ
Vin is he Coulomb in e ac ion be ween elec ons
ˆ
Vin =1
2
n
X
i=1
n
X
i6=j
e2
| i− j|,(1.7)
6
1 The Physical Sys em
and inally ENuc is he Coulomb in e ac ion be ween nuclei. I con ibu es o he
o al ene gy bu is no ele an o desc ibe he elec onic s a es.
ENuc =1
2
N
X
I=1
N
X
I6=J
ZIZJe2
|RI−RJ|(1.8)
Diagonalizing he elec onic Hamil onian o eq. (1.4) we can ob ain he ene gy
Es0({R})and he elec onic s a es ψs0({ };{R}).
The coe icien s χss0({R}) ha couple he nuclea -elec on sys em can be hough as
he nuclea wa e unc ions. They can be ob ained inse ing he exp ession (1.3) in o eq.
(1.1) and in eg a ing o e he elec on a iables { }leading o he ollowing equa ions
hˆ
TNuc +Es0({R})−Esi|χss0({R})i=−X
i0
Csi0|χsi0({R})i(1.9)
whe e ˆ
TNuc is he nuclea kine ic ene gy and he ma ix elemen s Csi0a e gi en by
Csi0=
N
X
J=1 me
MJhψs({ };{R})|∇J|ψi0({ };{R})i∇J
N
X
J=1 me
2MJhψs({ };{R})|∇2
J|ψi0({ };{R})i
(1.10)
we can apply now he adiaba ic app oxima ion by igno ing he o -diagonal e ms o
Csi0ma ix elemen s.2,3 Thus, he elec ons a e conside ed o emain in hei s a e s0
as he nuclei mo e. In spi e o he change in he ene gy Es0({R})and he wa e unc ion
ψs0({ };{R}), he s0s a e does no change and he nuclea mo ion does no cause any
elec onic exci a ion. The e o e, in he adiaba ic app oxima ion, he nuclea dynamics
a e desc ibed by a pu ely elec onic po en ial and he ull se o s a es sis a p oduc o
nuclea and elec onic s a es. I all he Csi0elemen s a e igno ed,4 he equa ion o he
nuclea mo ion simpli ies o
hˆ
TNuc +Es0({R})i|χn0s0({R})i=En0s0|χn0s0({R})i(1.11)
whe e n0he e labels nuclea s a es. The nuclea po en ial ene gy, ha we ha e
seen ha co esponds o he elec onic ene gy Es0({R}), is usually e med as he Bo n-
Oppenheime ene gy su ace. A ske ch o a po en ial ene gy su ace is shown in Fig. 1.2.
We can see he e se e al minima, hey co espond o di e en s uc u al phases in he
2In mos books he Bo n-Oppenheime and adiaba ic app oxima ions a e e e ed as he same ap-
p oach e en hough we ha e seen hey a e no s ic ly he same.
3O -diagonal e ms a e esponsible o he elec on-phonon coupling.
4Diagonal e ms a e neglec ed o he calcula ion o phonon ene gies in he adiaba ic app oxima ion
in he ozen-phonon and pe u ba ion me hods ha we will see below.
7
Adol o O e o Fumega
se o possible nuclea con igu a ions {R}. In p inciple, he g ound s a e o he sys em
will be he con igu a ion {R0} ha co esponds o he absolu e minimum o he po en ial
ene gy su ace. Phase ansi ions be ween di e en s uc u al phases can be d i en by
ex e nal ields such as p essu e, elec omagne ic ields o empe a u e.
Figu e 1.2: Ske ch o a Bo n-Oppenheime ene gy su ace Es0({R}). Di e en minima
ha e been d awn, hey co espond o di e en s uc u al phases in he se o possible
nuclea con igu a ions {R}. Phase ansi ions be ween hese phases can be d i en by
ex e nal ields such as p essu e, elec omagne ic ields o empe a u e.
The g ound s a e con igu a ion {R0}, i.e., whe e he nuclei a e loca ed in he g ound
s a e, can be compu ed using he o ce o Hellmann-Feynman heo em [10]. I allows o
compu e he nuclea o ces as
FI=−∂Es0
∂RI
=−hψs0|∂ˆ
He
∂RI|ψs0i,(1.12)
whe e RIis he posi ion o nucleus Iand FIis he o ce on i . In he g ound s a e
con igu a ion he o ces on all a oms mus be ze o, since he {R0}con igu a ion is a
minimum o he po en ial ene gy su ace.
The o ce heo em can be gene alized o any pa ame e λo he Hamil onian as
∂Es0
∂λ =hψλ|∂ˆ
He
∂λ |ψλi,(1.13)
and a ini e ene gy di e ence be ween wo di e en λs a es, conside ed o be g ound
s a es o each λ alue, can be ob ained as an in eg al o e a con inuous a ia ion o he
Hamil onian om one s a e o he o he . This is called he adiaba ic connec ion [11].
Finally, no e ha he adiaba ic app oxima ion ha we ha e conside ed in his sec ion
is well jus i ied as long as he gap in he elec onic exci a ion spec um is la ge han he
ene gies in ol ed in he nuclea dynamics. In me als o sys ems whe e he elec onic
s a es a e degene a e he app oxima ion migh ail.
8
1 The Physical Sys em
In his sec ion we ha e spli he many-body p oblem in o wo: on one side elec ons
and on he o he nuclei. We will see in he ollowing wo sec ions how o deal wi h each
o hem sepa a ely.
1.3 In e ac ing Elec ons
The Bo n-Oppenheime and adiaba ic app oxima ions le us wi h an elec onic p ob-
lem o sol e. The in e ac ing elec on Hamil onian ha desc ibes his sys em is gi en by
eq. (1.4). The o al ene gy Eo he elec onic sys em is he expec a ion alue o ha
Hamil onian ˆ
He
E=hΨ|ˆ
He|Ψi
hΨ|Ψi,(1.14)
whe e Ψis he many-body elec onic wa e unc ion Ψ({ };{R}). A his poin , we
s op using explici no a ion o he dependence on he nuclea coo dina es {R}since hey
a e jus pa ame e s in he elec onic p oblem.
I we conside Ψas a a iable ial unc ion, he eigens a es o he Hamil onian co e-
spond o he s a iona y poin s o he ene gy E. As we ha e seen in he i s sec ion o his
chap e , he eigen alues and eigens a es o he Hamil onian can be ob ained by sol ing
he ime-independen Sch ödinge ’s equa ion (eq. (1.1)). The g ound s a e wa e unc ion
Ψ0will be he s a e wi h he lowes ene gy. We can ob ain i by minimizing he o al
ene gy wi h espec o all a iables { }in Ψ. All he g ound s a e p ope ies will be
de e mined by he g ound s a e wa e unc ion Ψ0. The e o e, in o de o cha ac e ize he
elec onic s uc u e o a ma e ial, a leas i s in insic elec onic p ope ies, we need o
ob ain he g ound s a e wa e unc ion.5
A widely known me hod o ob ain he g ound s a e wa e unc ion Ψ0is he Ha ee-
Fock me hod [12]. This app oach assumes ha he many-elec on wa e unc ion Ψ({ })
can be exp essed as an an isymme ized de e minan al wa e unc ion o a ixed numbe n
o elec ons. The equi emen o be an isymme ic comes om he ac ha elec ons a e
e mions. In case ha no spin-o bi in e ac ion is included in he elec onic Hamil onian
ˆ
He, he wa e unc ion akes he o m o a Sla e de e minan [13]
Ψ({ }) = 1
(n!)1/2
φ1( 1)φ1( 2). . . φ1( n)
φ2( 1)φ2( 2). . . φ2( n)
.
.
..
.
.....
.
.
φn( 1)φn( 2). . . φn( n)
(1.15)
5The o he s a iona y poin s in he ene gy wi h espec o Ψco espond o exci ed s a es. In his
hesis we will ocus on he g ound s a e and small pe u ba ions o i , such as he ones p oduced by small
nuclei displacemen s.
9
Adol o O e o Fumega
ELDA
xc [ρ] = Zd3 ρ( )εhom
xc (ρ( )).(1.31)
The exchange ene gy o a homogeneous elec on gas as a unc ion o he densi y
can be compu ed analy ically. On he o he hand, he co ela ion e m has been
compu ed wi h g ea accu acy wi h quan um Mon e Ca lo me hods. The LDA
exchange-co ela ion unc ional p o ides in gene al a good app oxima ion o solids.
I gi es bond-leng hs wi h an accu acy a ound he 1% e o . Thus, i is capable o
making highly accu a e s uc u al p edic ions. Howe e , he ene gy gaps end o be
unde es ima ed wi hin his app oach.
•Semilocal Gene alized G adien App oxima ion (GGA): his app oach is he nex
na u al s ep o he LDA. The g adien o he densi y ∇ρ( )is in oduced in he
exchange-co ela ion ene gy as
EGGA
xc [ρ] = Zd3 ρ( )εhom
xc (ρ( ),|∇ρ( )|).(1.32)
The e is some eedom in he way ha he g adien is inco po a ed, so di e en GGA
app oaches can be de ined. We will use he one gi en by Pe dew, Bu ke and Enze ho
(PBE) [19]. This app oach has been used in a huge numbe o sys ems o bench-
ma king mos o he codes ha we will use. I imp o es LDA o many s uc u al
p edic ions. Howe e , as happens wi h LDA, band gaps a e also unde es ima ed,
and some imes a me allic beha iou (no ene gy gap) is p edic ed o co ela ed
insula o s. This beha iou has been e med he band gap p oblem o LDA and
GGA.
•Localized-o bi al app oxima ion (LDA+U): localized o bi al app oaches ha e been
de eloped o o e come he band gap p oblem in sys ems whe e elec ons end o
show a localized beha iou , like ansi ion me al oxides and a e-ea h elemen s [20].
These sys ems ha e pa ially illed dand s a es in which elec on-elec on in e -
ac ions become s ong, and so does he co ela ion ene gy. One o such localized-
o bi al app oxima ions is he LDA+U, which is based on he LDA and includes he
co ela ion e m U gi en by a Hubba d model.8The me hod di ides he sys em in
wo kinds o elec ons:
–Delocalized elec ons ha a e ea ed wi h he usual LDA app oach.
8The Hubba d model is a widely used model Hamil onian in he ield o highly-co ela ed elec on
sys ems. I leads o use ul esul s o Mo insula o s, in which co ela ions play a undamen al ole o
explain he insula ing beha iou . I has also been used as he building block o ea mo e exo ic sys ems
like high- empe a u e supe conduc o s o spin-liquids.
16
1 The Physical Sys em
–Localized elec ons ha include an on-si e Coulomb in e ac ion wi h he o m
1
2UPi6=jninj, whe e niand nj ep esen o bi al illing. The o bi al ene gies i
o hese localized elec ons can be compu ed leading o
i=LDA
i+U1
2−ni(1.33)
whe e LDA
iis he LDA con ibu ion o he ene gy and he las e m ake in o
accoun he Coulomb in e ac ions. Thus, we ha e ha o occupied o bi als
ni= 1, and hence he ene gy o hose o bi als is
i=LDA
i−1
2U
and o unoccupied ni= 0
i=LDA
i+1
2U.
We can obse e ha he e ec o he U is o p oduce an ene gy shi be ween
he occupied and he emp y s a es. Thus, leading o an ene gy gap due o
elec on-elec on in e ac ions in Mo -Hubba d sys ems.
The U e m can be calcula ed by means o many-body me hods ha a e a beyond
he scope o he p esen hesis. The impo an poin o us is ha LDA+U imp o es
subs an ially (quan i a i e and quali a i e) LDA and GGA app oaches o sys ems
in which elec on localiza ion happens. Since his hesis is de o ed o he s udy o
ansi ion me al sys ems, we will ind in he LDA+U app oach a good desc ip ion
o many o such sys ems. I will upg ade band gaps, magne ic momen s o exchange
coupling cons an s.
•Modi ied Becke-Johnson po en ial (MBJ): his me hod is able o p o ide Kohn-
Sham gaps close o expe imen al alues. The po en ial is buil upon he Becke-
Johnson po en ial (BJ) [21], whose goal is o ep oduce he uni o m gas limi and
desc ibe shell s uc u es which a e cha ac e is ic o exac exchange po en ials. The
kine ic ene gy densi y is in oduced o ea co ec ly hyd ogenic sys ems. The BJ
po en ial was hen modi ied by T an and Blaha [22], and is he one ha we will use
in his hesis.
The MBJ po en ial is a semilocal po en ial, wha makes MBJ compu a ionally
cheape han non-local hyb id unc ionals. Howe e , i mus be no ed ha i is
a po en ial and no a unc ional, so i canno be used o minimize he o al ene gy.
I is employed along ano he unc ional such as LDA o GGA used o sol e he
Kohn-Sham p oblem. The densi y and eigen unc ions a e hen used o build he
MBJ po en ial and o compu e he desi ed band gaps wi h high accu acy.
17
Adol o O e o Fumega
•Van de Waals ( dW): hese non-local unc ionals a e designed o ea dispe si e
o ces ha a e o igina ed om he long- ange in e ac ions o quan um luc ua ing
elec ic dipoles. The local and semilocal LDA and GGA a e no able o desc ibe any
in e ac ion o such cha ac e is ics. In ou s udies on ansi ion me al compounds we
will deal wi h laye ed compounds, whose laye s a e sligh ly bonded ia an de Waals
in e ac ions. The e o e, we will need o use such a kind o non-local unc ionals o
co ec ly p edic hei s uc u al p ope ies. An al e na i e is o include an de
Waals in e ac ions in a semiempi ical way. This is done by adding a dispe sion
ene gy e m Edisp o he o al ene gy:
Edisp =−1
2
N
X
I=1
N
X
I6=J
C6IJ
|RI−RJ|6 d(|RI−RJ|)(1.34)
whe e he summa ions ange o e he numbe o a oms N,C6IJ is he dispe sion
coe icien o he a om pai IJ, and d(|RI−RJ|)is a damping unc ion ha a oids
di e gences a nea dis ances. I a ies smoo hly om 0 a close dis ances o 1 a
la ge ones.
Finally, some gene al ema ks abou exchange and co ela ion a e: i) he exchange
ene gy domina es o e co ela ion, since he main e ec o exchange is o emo e he spu-
ious sel -in e ac ion in oduced by he Ha ee ene gy, and ii) he inclusion o exchange
and co ela ion always lowe s he g ound s a e ene gy o he sys em. They in oduce wha
has been e med as he exchange-co ela ion hole, which is a dec ease in he p obabili y o
inding elec ons nea each o he due o he exclusion p inciple and he epulsi e Coulomb
in e ac ions.
1.3.3 Spin and Rela i is ic E ec s
So a we ha e no di ec ly alked abou spin. As we know, elec ons a e Di ac
e mions o spin 1/2. Spin a ises na u ally in he co a ian Di ac o mula ion o quan um
mechanics. The Di ac equa ion is linea in ime and momen a in o de o be Lo en z in-
a ian . Solu ions o his equa ion in ol e commu a ion equi emen s ul illed by ce ain
ma ices9and a ou -componen wa e unc ion, called spino . We see ha his new o mu-
la ion en ails mo e complexi y o he many-elec on p oblem ha we ha e been simpli y-
ing. As a esul , he app oach ha is aken is o con inue wo king wi h he Sch ödinge ’s
equa ion ha we ha e been using hus a , and include he ela i is ic e ec s ha come
om he Di ac equa ion on i . The elec onic wa e unc ion is hen exp essed in e ms o
a wo componen wa e unc ion:
Ψ( ) = Ψ↑( )
Ψ↓( )(1.35)
9The way in which hese ma ices a e chosen leads o desc ibe Di ac o Majo ana e mions.
18
1 The Physical Sys em
whe e Ψ↑( )and Ψ↓( ) ep esen each spin wa e unc ion along a chosen axis.10 New
e ms due o he magne ic na u e o spin and ela i is ic e ec s a e included in Sch ödinge ’s
equa ion:11
i~∂Ψ( )
∂ =ˆ
He=π2
2me
+V+ˆ
HZ+ˆ
HSOΨ( )(1.36)
whe e he e ec s o a magne ic ield Bin oduce he ec o po en ial Ain he mo-
men um ope a o pas π=p−(e/c)A. Also a Zeeman e m ˆ
HZ=µBσB, whe e
µB=e~/2mecis he Boh magne on. Finally, he spin-o bi in e ac ion ˆ
HSO, which can
be iewed as he in e ac ion o he elec on wi h momen um pand spin σmo ing in an
elec ic ield, o equi alen ly as a magne ic ield in he es ame o he elec on. This
e m is he only one ha mixes spin ↑and ↓componen s as
ˆ
HSO =ˆ
HSO(↑↑)ˆ
HSO(↓↑)
ˆ
HSO(↑↓)ˆ
HSO(↓↓)(1.37)
Equa ion (1.36) p o ides a way o in oduce spin-o bi and magne ic e ec s in he
non-in e ac ing elec on Hamil onian. The e o e, hey can be included in he Kohn-
Sham DFT leading o a spin-DFT. The inclusion o a magne ic ield in DFT is no
i ial, i would equi e going o a densi y-cu en unc ional heo y o ea co ec ly
he i s e m in eq. (1.36). This e m will be esponsible o he e ec o he magne ic
ield in he o bi al mo ion.12 Howe e , he Zeeman e m can be easily included, since i
a ec s only he spin, leading o an ene gy shi be ween bo h spin channels. Thus he
e ec i e po en ial ha appea s in he Kohn-Sham equa ion will ha e om now on a spin
σdependency Vσ
e . The e o e, we will ha e wo di e en spin channels o he elec on
densi y ρ( ) = ρ↑( ) + ρ↓( ).
In he absence o an ex e nal Zeeman ield, i may occu ha he lowes -ene gy
solu ion is spin pola ized, and hence ρ↑( )6=ρ↓( ), hus, b eaking he spin degene acy
and leading o a magne ic sys em.
In ou calcula ions, he spin-o bi in e ac ion will be in oduced in a second- a ia ional
manne using he scala ela i is ic app oach. This app oxima ion is de i ed when spin-
o bi is ea ed as a pe u ba i e e ec [23].
1.3.4 Elec onic S uc u e Me hods
In his pa we will see di e en implemen a ions o sol e he Kohn-Sham equa ions
ha we ha e been desc ibing abo e. All he me hods ha we will see ag ee when used
10In he calcula ions p esen ed in his hesis we will conside ha spin is quan ized along a gi en axis.
This simpli ies eno mously he ea men o spin e ec s. Howe e , i mus be no ed ha o he s udy o
ce ain sys ems, like he ones ha p esen spin ex u es, a noncollinea ea men o spin is equi ed.
11Spin-independen scala e ms (Da win and mass- eloci y) a e no included.
12In gene al, we will no deal wi h he inclusion o magne ic ields in his hesis. The only s udy in
which we ha e included a magne ic ield is he one in Chap e 10. The e ec on he o bi al mo ion is
included he e h ough a Peie ls subs i u ion in a igh -binding Hamil onian.
19
Adol o O e o Fumega
wi h enough ca e and con e gence [24]. Howe e , each o hem has i s ad an ages and
in icacies ha mus be conside ed when ackling a pa icula p oblem. The me hods can
be di ided by he choice o basis se used o expand he Kohn-Sham o bi als and also by
he numbe o elec ons and po en ial conside ed.
A na u al basis se o expand he Kohn-Sham o bi als in c ys als is he plane-wa e
basis se . C ys als a e he kind o sys em ha we will s udy in his hesis and, as we
will see in he nex Chap e , ha e a spa ial pe iodici y ha can be exploi ed by he
use o plane wa es. Plane wa es a e eigen unc ions o he Sch ödinge equa ion wi h
cons an po en ial, so hey a e an excellen choice o sys ems in a nea ly- ee-elec on
egime. Howe e , nea he nuclei elec ons end o be localized, so hei wa e unc ions
a y apidly. They a e usually called co e elec ons. The use o plane wa es o desc ibe
hese wa e unc ions becomes inapp op ia e, since an eno mous amoun o plane wa es
would be equi ed o expand such localized co e wa e unc ions. Thus, he me hods ha
we p esen below co espond o he ones implemen ed in he codes ha we will use. They
basically show di e en ways o deal wi h co e elec ons.
The APW+lo me hod
The augmen ed plane wa e plus local o bi als (APW+lo) is one o he mos accu a e
and e icien me hods o sol e he Kohn-Sham equa ions [25, 26]. The me hod is based
on he o iginal augmen ed plane wa e me hod APW by Sla e in 1937 [27]. The idea is o
di ide he space in wo egions as shown in Fig. 1.4. We will ha e he mu in- in sphe es
cen e ed on he a oms13 and an in e s i ial egion be ween hem. Di e en basis unc ions
a e used in each egion:
ϕ( ) = (Pncneikn i ∈In e s i ial egion
Plm Almul( , El)Ylm i ∈Mu in- in sphe e (1.38)
So we see ha he in e s i ial egion is expanded by plane wa es, while wi hin he
mu in- in sphe es a combina ion o adial unc ions ul( , El) imes sphe ical ha monics
Ylm is used, since hey a e solu ions o he adial Sch ödinge equa ions. The coe icien s
o he expansion a e cnand Alm. These mus ul ill ce ain equi emen s in e ms o
c ys al pe iodici y and bounda y ma ching o he mu in- in and in e s i ial egions.
In he o iginal APW me hod he adial basis se was ene gy dependen ul( , El), which
induces undesi ed non-linea i ies in he Kohn-Sham p oblem. Di e en schemes we e
p oposed o sol e his issue like he linea ized augmen ed plane wa e me hod (LAPW)
[28, 29]. In he APW+lo, he ene gy Elo he adial unc ions ul( , El)is ixed in o de
o keep linea he p oblem. This educes he a ia ional lexibili y o he adial basis se .
In o de o eco e i , local o bi als ϕlo( )a e included wi hin he mu in- in egion:
ϕlo( ) = [almul( , E1l) + blm ˙ul( , E1l)] Ylm (1.39)
13The name mu in- in sphe es comes om hei simila shape o ki chen molds o mu ins.
20
1 The Physical Sys em
Figu e 1.4: Schema ic ep esen a ion o he space di ision in e ms o mu in- in sphe es
a ound he a oms and he in e s i ial egion be ween hem.
whe e alm and blm a e coe icien s o he expansion and E1la e he linea iza ion
ene gies whe e he solu ions o he adial Sch ödinge equa ion a e e alua ed. These local
o bi als a e equi ed o be ze o a he sphe e bounda y as well as no malized.
The po en ial is expanded in he APW+lo me hod as
V( ) = (PkVkeik i ∈In e s i ial egion
Plm Vlm( )Ylm i ∈Mu in- in sphe e (1.40)
and he cha ge densi y in an analogous way. So, no app oxima ions a e made o i s
shape, which is he eason why his me hod is a ull-po en ial all-elec on me hod. Thus,
compa ed o o he me hods is by a he mos e icien and accu a e.
The con e gence o he APW+lo basis se is con olled by he pa ame e Rm Kmax,
whe e Rm is he smalles a omic sphe e adius and Kmax is he alue o he la ges kn
ec o in he expansion gi en by eq. (1.38). Values o Rm Kmax ∼7p o ide a good
con e gence o he ansi ion me als ha we ha e s udied in his hesis.
The APW+lo me hod is implemen ed in he wien2k code [30] ha we will use in
mos o he s udies p esen ed in he second pa .
Pseudopo en ials and he PAW me hod
As we ha e men ioned, ou aim is o use plane wa es as he basis se . Howe e ,
he basis se becomes oo la ge o localized elec ons nea he nuclei, i.e., o he co e
elec ons. Thus, making plane wa es imp ac ical.
The idea o pseudopo en ials was i s p oposed by Hans Hellman in 1934 [31]. This
app oach is based on he ac ha in a ma e ial, he elec ons ha a e going o play a
ole in i s p ope ies a e he alence elec ons, while he co e elec ons o a speci ic a om
emain p ac ically una ec ed by he o he a oms. The bonding be ween a oms will be
de e mined by he alence elec ons, and hence hey will be he impo an elec ons o
desc ibe.
21
Adol o O e o Fumega
The pseudopo en ial app oach p oposes o build an e ec i e po en ial and eplace
wi h i he a omic all-elec on ull-po en ial elimina ing he co e elec ons. Valence elec-
ons a e hen desc ibed by pseudowa e unc ions ϕi,ps ha oscilla e much less, i.e., hey
ha e less nodes, and hence i is possible o desc ibe hem wi h a easonable plane-wa e ba-
sis se . A schema ic o he pseudopo en ial idea is shown in Fig. 1.5. No e he smoo hness
o he pseudowa e unc ion which allows o desc ibe i wi h ewe plane wa es.
Figu e 1.5: Schema ic o a pseudopo en ial and pseudowa e unc ion. A smalle plane-
wa e basis se is equi ed o desc ibe he pseudowa e unc ions due o hei less oscilla ing
beha iou . The pseudowa e unc ions and pseudopo en ial ma ch he eal ones ou side
he cu -o adius.
Pseudopo en ials a e usually cons uc ed om ab in io a omic calcula ions. An
a omic e e ence s a e is conside ed by de ining a cu -o adius Rc o he co e. Ou -
side Rc, he pseudowa e unc ions and he eal wa e unc ions o he alence elec ons a e
equi ed o ha e he same ene gies and ampli ude (densi y). So pseudopo en ials a e
hose ha ha e a la ge cu -o adius. This helps o educe he size o he plane wa e
basis se needed o desc ibe he pseudowa e unc ions. Howe e , hese pseudopo en ials
a e less ans e able, i.e., accu acy is los depending on he chemical en i onmen . Thus,
pseudopo en ials deal wi h a comp omise be ween ans e abili y and basis-se size. The
so-called small co e app oxima ion is aken by assuming ha he e is no o e lap be ween
co e and alence s a es. P oblems migh a ise when dealing wi h semico e elec ons.
The mos common o ms o pseudopo en ials a e he no m-conse ing and ul aso
pseudopo en ials [32, 33]. In he case o no m-conse ing pseudopo en ials, he no m o
he pseudowa e un ions and he eal Kohn-Sham wa e un ion a e equi ed o be he same
inside Rc. Ul aso pseudopo en ials elax his cons ain , which educes he size o he
22
1 The Physical Sys em
equi ed basis se . In e u n, a gene alized eigen alue p oblem is in oduced. I we de ine
he non-ze o no m di e ence ñij inside Rcas:
ñij =hϕi|ϕji−hϕi,ps|ϕj,psi,(1.41)
so now he no malized s a e o he pseudo Hamil onian ˆ
Hps obeys he equa ion:
ˆ
Hpsϕi=iˆ
Tupsϕi,ps,(1.42)
wi h he ope a o ˆ
Tups being a ans o ma ion
ˆ
Tups =1+X
<Rc,ij |piiñij hpj|,(1.43)
whe e pia e p ojec o s ha o m a basis wi h he pseudo e e ence s a e inside Rc.
We ha e seen ha he idea o pseudopo en ials en ails neglec ing he co e elec ons.
Howe e , based on he p ojec o s in oduced o ul aso pseudopo en ials and he no ion
o augmen a ion explained o he APW me hod, co e s a es can be somehow eco e ed
wi hin he P ojec o Augmen ed Wa e (PAW) me hod [34]. The idea is o in oduce a
ans o ma ion ˆ
TPAW simila o he one in eq. (1.43) o ul aso pseudopo en ials. How-
e e , in his new ans oma ion, he ull-elec on Kohn-Sham wa e unc ions ϕicalcula ed
in he a omic e e ence sys em a e s ill in ol ed:
ˆ
TPAW =1+X
i
[|ϕii−|ϕi,psi]hpi|.(1.44)
The PAW me hod is usually combined wi h he ozen co e app oxima ion, whe e
co e wa e unc ions a e conside ed o be una ec ed by he en i onmen . The e o e, hey
a e no modi ied h ough he calcula ion.
Pseudopo en ials and he PAW me hod a e implemen ed in he quan um esp esso
[35] and VASP [36] codes. We will make use o hem in he s udies shown in he second
pa .
A clea ad an age o he pseudopo en ials and PAW me hod is ha hey a e much
as e han he APW+lo me hod and he e o e ma e ials wi h mo e a oms can be ea ed
mo e easily. Fo ces a e also implemen ed in an easie way. This is he eason why we
will use many imes pseudopo en ials o s uc u al elaxa ions and ela ed p ope ies. In
con as , he APW+lo me hod is he mos accu a e me hod o sol ing he Kohn-Sham
equa ions. Analysis o he elec onic s uc u e and magne ic p ope ies o ansi ion
me al compounds equi es a good desc ip ion o he d elec ons. These elec ons migh
be conside ed as semico e s a es when hey appea localized a low enough ene gies. The
pseudopo en ial me hod could be sca ce in his case, meanwhile he APW+lo me hod
ensu es a ai ea men o he d elec ons.
23
Adol o O e o Fumega
1.3.5 Simpli ied Models
In his inal subsec ion we will see ha i is possible o wo k wi h simple models ha
desc ibe he essen ials o a pa icula elec onic s uc u e. These models can be buil om
heo e ical assump ions o can be dis illed om ab ini io calcula ions.
We will discuss wo simple models: he so-called igh -binding and Heisenbe g models.
They a e p obably he easies models ha we can ind. Howe e , hey allow o gain insigh
in o he physical p oblem and o compu e ce ain p ope ies ha o he wise could no be
calcula ed.
Tigh -Binding Model
The i s igh -binding (TB) model was in oduced by Bloch in 1928 [37]. He consid-
e ed a simple s-symme y unc ion as he basis o i . In 1934 Jones, Mo and Skinne
1934, ex ended he model o a basis o di e en a omic o bi als [38].
The idea o he TB model is o ep esen he elec onic wa e unc ions as localized
o bi als ai( −RI), each associa ed wi h an a om a posi ion RI, whe e ilabels all he
s a es in he basis. Hence he ma ix elemen s o he Hamil onian Hij o a pai o s a es
iand jcen e ed a RIand RJ espec i ely, a e:
Hij =Zd ai( −RI)ˆ
Haj( −RJ),(1.45)
and he o e lap be ween he s a es Sij is:
Sij =Zd ai( −RI)aj( −RJ).(1.46)
Thus, he p oblem is ep esen ed in a ma ix o m, whe e he basis se does no need
o be explici ly speci ied. Only he ma ix elemen s o he o e lap and he Hamil onian
a e equi ed o sol e he elec onic s uc u e p oblem. No hing bu a ma ix diagonal-
iza ion is needed o ob ain he eigen alues o he ene gy and he eigen unc ions as linea
combina ions o a omic o bi als.
A omic o bi als a e usually aken as he localized basis, since he di e en bonding
ypes be ween hem a e well known.14 In Fig. 1.6 σand πbondings be ween s and p
o bi als a e depic ed. In 1954 Sla e and Kos e se he undamen als o he TB me hod,
whe e hey included he celeb a ed Sla e -Kos e ables o he in e a omic ma ix el-
emen s [39]. O bi al and ansla ional symme ies a e conside ed o de i e he ma ix
elemen s ha appea on hose ables.
The eal space app oach p o ided by a TB Hamil onian enables o compu e hings
like su ace s a es o non- i ial opological ma e ials, o o include he e ec o a magne ic
ield in he o bi als. Scalabili y is ano he ad an age o his kind o models, hus, allowing
o ea sys ems wi h a huge numbe o a oms.
14O he basis se s like he ones ob ained om a Wannie iza ion a e commonly used.
24
1 The Physical Sys em
Figu e 1.6: Schema ic ep esen a ion o he di e en σand πbondings be ween s and p
a omic o bi als used o buil a igh -binding model.
The TB Hamil onian can be o mula ed in he ame o second quan iza ion leading
o a summa ion o kine ic and po en ial ene gy e ms:
ˆ
H=X
i
Eia†
iai+X
i6=j
ij ha†
iaj+a†
jaii,(1.47)
whe e Eiis he so-called onsi e ene gy o a gi en o bi al, i.e., he sum o he kine ic
and po en ial ene gies o o bi al i, and he ij is he so-called hopping e m be ween he
o bi al iand j. I ep esen s he kine ic ene gy an elec on needs o hop om one o bi al
o he o he . Mo e complex e ms could be included in he simple TB Hamil onian gi en
by eq. (1.47). Those new e ms would lead o e y di e en models like he Hubba d
model ha can be used o explain me al-insula o ansi ions, o many o he s o explain
mo e exo ic phenomena like supe conduc i i y o opological insula o s.
We will wo k wi h TB Hamil onians on he second pa . The Py hTB [40] and
Pybinding [41] packages will be used o build TB Hamil onians and o compu e di e en
p ope ies.
Heisenbe g Model
Finally, we p esen he Heisenbe g model. This is he mos ep esen a i e example o
a spin Hamil onian. In sys ems ha show a magne ic insula o beha iou i is possible
o ge id o he cha ge deg ees o eedom and build an e ec i e spin Hamil onian. The
insula o beha iou could also be elaxed when he spin Hamil onian is buil o localized
s a es, like he d elec ons ound in he ansi ion me al compounds ha we analyze in
he second pa . Thus, he sys em is simpli ied o a spin model, whe e spins lay localized
a a gi en si e iand in e ac be ween hem. A schema ic o he model is depic ed in Fig.
1.7. In he case o he Heisenbe g model, con inuous o a ions o spins a e allowed. The
Hamil onian is gi en by:
25
Adol o O e o Fumega
linea esponse o a single pe u ba ion is o he same o de as he one o he unpe u bed
densi y.
One pa icula ad an age o DFPT o compu e he ha monic phonon spec a com-
pa ed o o he me hods is ha he esponse o pe u ba ions o di e en wa eleng hs is
decoupled. In e u n i mus be no ed ha i is a pe u ba i e me hod, which en ails
ha he app oxima ion b eaks o high-ene gy anha monic co ec ions o he phonons o
when dealing wi h uns able s uc u es.
DFTP is implemen ed in he quan um esp esso and VASP codes. We will make
used o i in he s udies o he second pa o compu e ha monic p ope ies.
F ozen Phonon me hod
A di e en app oach o ob ain he coe icien s o he Taylo expansion o he po en ial
ene gy su ace is he so-called ozen-phonon me hod o ini e-displacemen app oach [47].
The idea o his me hod is e y simple, a se o s uc u es is gene a ed whe e di e en
a omic displacemen s ∆uIα a e pe o med. The in e a omic o ce cons an s DIα,I0α0a e
hen app oxima ed as:
DIα,I0α0≈ −FI0α0+∆uIα −FI0α0
∆uIα
(1.69)
Thus, o he se o s uc u es wi h he displacemen s we will ha e o compu e he
a omic o ces FI0α0+∆uIα wi h espec o he undis o ed s uc u e. Usually FI0α0= 0 since
he undis o ed s uc u e is he one whe e he a omic posi ions a e in equilib ium.
The in e a omic o ce cons an s can be ob ained om eq. (1.69); he Pa linski-Li-
Kawazoe me hod [48], which is jus a nume ical i ing app oach o ob ain he o ce
cons an s om a se o o ces and displacemen s.
The e o e, only he compu a ion o he o ces om a se o s uc u es wi h di e en
displacemen s is equi ed. The size o he se can be cle e ly limi ed by in oducing he
symme ies o he s uc u e. An analogous p ocedu e can be de eloped o ob ain he
hi d-o de o ce cons an s.
One ad an age o his me hod is ha i allows o include anha monic e ec s ha
would be di icul o s udy h ough DFPT. Mo eo e , i can be implemen ed in any
s anda d DFT code able o compu e o al ene gies and o ces. Howe e , he ozen-phonon
app oach is much mo e compu a ionally demanding, in he case o c ys als i migh equi e
una o dable supe cells o compu e he phonon spec a a a desi ed wa eleng h ec o .
The ozen-phonon app oach o compu e he second and hi d-o de o ce cons an s
is implemen ed in he Phonopy [49] and Phono3py [50] packages. We will use hem in
he s udies o Pa II.
32
1 The Physical Sys em
Ab ini io Molecula Dynamics and he Tempe a u e Dependence E ec i e
Po en ial me hod
Finally, we will b ie ly p esen how o include he e ec o ini e empe a u e in a
phonon calcula ion ia he empe a u e e ec i e po en ial (TDEP) me hod [51].
Un il now, we ha e been wo king wi h he po en ial ene gy su ace o Bo n-Oppenheime
ene gy su ace. The ene gy minimum o such su ace gi es he equilib ium posi ions o he
nuclei. This is app op ia e when he e ec o empe a u e can be neglec ed, i.e., i he e
a e no empe a u e-dependen dynamical ins abili ies o s ong anha monic e ec s ha
lead o s uc u al ansi ions. In such cases, whe e ini e empe a u e o p essu e is in-
oduced, he po en ial ene gy su ace is no sui able, and he Gibbs ee ene gy becomes
he p ope he modynamic quan i y o minimize.
We will be in e es ed in he si ua ion whe e a s uc u al phase ansi ion occu s a a
ansi ion empe a u e Ts, i.e., equilib ium posi ions {R0}will be di e en in each phase.
Thus, we will ha e wo di e en egimes. On one hand, he low empe a u e one ha can
be analyzed by he me hods ha we ha e been desc ibing o he po en ial ene gy su ace.
On he o he hand, he high empe a u e phase ha equi es a di e en ea men .
Figu e 1.8: Schema ic ep esen a ion o he ee ene gy su ace as a unc ion o empe a-
u e. Equilib ium posi ions {R0}({R0
0}) a e ep esen ed in he low (high) empe a u e
phase T < Ts(T > Ts) as he minimum o he solid (dashed) line.
The TDEP is based on he idea o gene a ing a se o he malized s uc u es a a ini e
empe a u e and ex ac ing he o ce cons an s o a model Hamil onian ha ma ches he
con igu a ions. The way o explo e he phase space a a gi en empe a u e is by means
o an ab ini io molecula dynamics (aiMD) simula ion in he canonical ensemble. A a
ce ain numbe o ime s eps he sys em will be equilib a ed a he chosen empe a u e.
Then, he in e a omic o ce cons an s a a gi en o de can be ex ac ed by minimizing
33
Adol o O e o Fumega
he di e ence in he o ces ob ained in he aiMD and hose om he model Hamil onian
in oduced by he TDEP me hod.18
We end his chap e in which we ha e desc ibed how o ackle he many-body
p oblem o nuclei and elec ons. The Bo n-Oppenheime and adiaba ic app oxi-
ma ions allowed o spli he analysis in nuclei and elec ons. Densi y unc ional
heo y was desc ibed o he ea men o elec ons and la ice dynamics o nuclei.
Thei p ac ical implemen a ion was p esen ed along wi h me hods o sol e hem.
All he app oxima ions and he me hods ha we ha e been desc ibing so a a e
summa ized in Fig. 1.1.
18In an aiMD a omic posi ions a e mo ed h ough a pe iod o ime. A small ime s ep is de ined,
a DFT calcula ion is pe o med a each o hese s eps o de e mine he o ces and p oduce he a omic
displacemen s ha lead om one s ep o he nex one. Thus, p oducing he dynamics. In he canonical
ensemble he se o possible con igu a ions is es ic ed by he gi en empe a u e. The way o in oduce
i in he calcula ions is by means o a hea ba h ha en e s he Hamil onian cons aining he phase space.
Di e en he mos a s can be ound.
34
2 Symme y
“Beau y is a he a ligh ha plays
o e he symme y o hings han
ha symme y i sel ."
— Plo inus
In his chap e we will alk abou he concep o symme y and how i is applied o
he many-body p oblem. Symme y is p obably he mos undamen al idea in Physics.
Acco ding o Emmy Noe he ’s heo em, e e y conse a ion law is a consequence o he
p esence o a pa icula symme y in he sys em [52]. Fo ins ance, ansla ional sym-
me y o space en ails linea momen um conse a ion, o angula momen um is ela ed
o o a ional in a iance. In a quan um sys em, he eigen alues’ degene acy o he eigen-
unc ions’ p ope ies a e de e mined by symme y, and hei ans o ma ion due o phase
ansi ions can be unde s ood as changes in he symme y. Mo e han ha , many o
he me hods ha we ha e al eady been desc ibing in Chap e 1, a e based on symme-
y conside a ions, such us he basis se s needed o desc ibe co e and alence elec ons.
The e o e, such a undamen al concep dese es a close look.
The aim o his chap e will be o p o ide he basics o analyze a sys em o nuclei
and elec ons h ough he eyes o symme y g oups. We will see in he second pa how
symme y guides all he analysis ha we pe o m.
2.1 De ini ion o a Symme y G oup
The way in which symme y en e s in he ma hema ical desc ip ion o a sys em is
h ough ans o ma ions. I a ans o ma ion Uis pe o med o he wa e unc ion Ψ ha
codi ies he s a e o a quan um sys em as
UΨ = Ψ,(2.1)
and we ob ain ha he ans o ma ion keeps he sys em in he same s a e Ψ, we say
ha he sys em is in a ian unde such ans o ma ion U.
A se o ans o ma ions {U} o ms a g oup i all i s elemen s ul ill he ollowing ou
equi emen s:
35
Adol o O e o Fumega
•The p oduc o any wo elemen s o he g oup is i sel an elemen o he g oup.
•The associa i e law applies.
•The uni (o iden i y) elemen Eis p esen in any g oup and i s p oduc wi h any
elemen o he g oup lea es he elemen unal e ed.
•Fo e e y elemen o he g oup he e is an in e se elemen such ha he p oduc o
he elemen and i s in e se leads o he iden i y elemen .
The g oup will be a symme y g oup i i s elemen s lea e he sys em in a ian as
gi en by eq. (2.1). Mo eo e , he g oup will be called Abelian i all he elemen s o he
g oup commu e. In gene al, his case does no happen. Non-Abelian g oups a e an ac i e
a ea o esea ch due o hei p omising applica ions.
We ha e s a ed in he p e ious chap e ha he Hamil onian o a sys em desc ibes all
he in e ac ions be ween nuclei and elec ons. The g ound s a e o he sys em is ob ained
as he lowes ene gy eigens a e o he Hamil onian. Fo his eason, he symme y analysis
is usually pe o med on he Hamil onian. I a sys em is in a ian unde a ans o ma ion
o a gi en symme y g oup, he Hamil onian mus commu e wi h he elemen s o ha
g oup. Fo ins ance, o he case o eigens a es wi h he same ene gy (degene a e), we
can ob ain one o he eigen unc ions and hen ob ain he o he s by applying o i he
symme y ope a ions ha lea e he Hamil onian in a ian .
Since his hesis is ocused on he analysis o c ys alline ansi ion me al compounds,
in he ollowing sec ions we will analyze he kind o ans o ma ions ha a e encoun-
e ed in hose sys ems. Such ans o ma ions de ine he di e en symme y g oups ha
cha ac e ize he p ope ies o he c ys al.
2.2 Poin G oup Symme y
Ro a ions, in e sions, e lec ions and hei di e en combina ions ha lea e a sys em
in a ian a e called poin symme ies. The ope a ion o a poin ans o ma ion Rpon a
unc ion ( )(such as he densi y o a sys em) is:
Rp ( ) = (Rp ).(2.2)
I a sys em has in e sion symme y, i means ha i is in a ian unde he change o
sign o all he coo dina es, o ins ance (x, y, z) = (−x, −y, −z). In he case o mi o
symme y (o e lec ion), he sys em is in a ian unde sign in e sion wi h espec o he
mi o plane. Fo example, i he mi o plane is he xy plane, he sys em is in a ian as
(x, y, z) = (x, y, −z).
Le us ocus now on he case o o a ions. Fo a ee a om, all o a ions abou any
axis a e symme y ope a ions, i.e., hey commu e wi h he Hamil onian. Thus, he ee
a om has ull o a ion symme y. The basis unc ions o he ull o a ion g oup a e he
36
2 Symme y
sphe ical ha monics Ylm. This is he eason why he a omic o bi als s,p,dand so on,
a e exp essed as a p oduc o a adial unc ion Rnl and a linea combina ion o sphe ical
ha monics. Thus, o he so bi als we ha e
s=Rn0Y00,(2.3)
o he po bi als
pz=Rn1Y10
px=Rn1
√2(Y11 +Y1−1)
py=Rn1
i√2(Y11 −Y1−1),
(2.4)
and o he do bi als
dz2=Rn2Y20
dx2−y2=Rn2
√2(Y22 +Y2−2)
dxy =Rn2
i√2(Y22 −Y2−2)
dxz =Rn2
√2(Y21 −Y2−1)
dxy =Rn2
i√2(Y21 +Y2−1),
(2.5)
whe e he subsc ip nhe e deno es he p incipal quan um numbe .
We see ha hese o bi als expand he elec onic s a es o a ee a om. Mo e han
ha , co e elec ons ha dly eel he p esence o neighbou ing a oms in a ma e ial, i.e., he
in e ac ions o co e elec ons wi h he elec ons o neighbou ing a oms a e iny. Fo his
eason, a omic-like o bi als a e a good basis o desc ibe hem. This is he idea behind he
APW me hod explained in he p e ious chap e , whe e co e elec ons inside he mu in- in
sphe e a e expanded as a p oduc o adial unc ions and sphe ical ha monics (eq. (1.38)).
In he ee a om (o nea ly ee a om, like he co e s a es ha we men ion), he
ene gy le els o a gi en nand lquan um numbe s a e degene a e. Fo ins ance, he
i e 3do bi als a e degene a e in ene gy.1This si ua ion changes when b eaking he ull
o a ional symme y by placing he ee a om wi hin a c ys al. The en i onmen c ea ed
by he neighbou ing a oms b eaks he ull o a ional symme y, hus, d i ing he in ini e
symme y g oup o a ini e one, in which only o a ions o a ce ain angle lea e he sys em
in a ian .
1A his poin we a e no conside ing spin degene acy explici ly. Howe e , no e ha o he 3do bi al
we ha e 10 degene a e ene gy le els i we coun spin.
37
Adol o O e o Fumega
Figu e 2.1: C ys al- ield spli ings o he d-o bi al ene gy le els in di e en en i onmen s.
The cen al a om is depic ed in blue and ligands in ed.
38
2 Symme y
The ini e o a ion g oups ha appea by he placemen o nuclei in a c ys al b eak he
degene acy acco ding o hei symme y ope a ions. This e ec is wha is usually known
as c ys al- ield spli ing o he o bi als. Since ou aim is o s udy c ys alline ansi ion
me al compounds, in Fig. 2.1 we summa ize he c ys al- ield spli ing on do bi als o
di e en kinds o en i onmen s (ligand a oms su ounding a cen al a om). Oc ahed al,
squa e plana , e ahed al en i onmen s and di e en dis o ions o hem a e analyzed
he e. We show how he ene gy le els co esponding o he do bi als o he cen al a om
a e spli .
O cou se, in he case o so bi als he e is no c ys al- ield spli ing, since he so bi al
is non-degene a e. The po bi als, which a e iply degene a e in he ee a om, migh be
spli by a c ys al ield. These spli ings can be explained in e ms o spa ial aniso opies.
I he e is no anyso opy, he h ee po bi als emain degene a e. I he e is aniso opy
along one axis, he o bi al ha co esponds o ha axis ge s spli om he o he wo.
Finally, i he h ee axes a e inequi alen , all po bi als become non-degene a e.
We ha e been analyzing poin g oup symme y in e ms o he elec onic wa e unc-
ions. Howe e , poin g oup symme y a ec s o , be e o say, cons ains he ene gy
le els o he phonon modes oo. The pola iza ion ec o s o he modes can be ana-
lyzed in an analogous way o he elec onic s a es by poin g oup symme y. E en mo e,
aniso opies in mac oscopic p ope ies such as conduc i i y a e also ela ed o he poin
g oup symme ies o he nanos uc u e.
2.3 T ansla ional Symme y
In his hesis we a e conce ned wi h he s udy o c ys als, in which ansla ional
symme y plays a undamen al ole. C ys als a e an o de ed s a e o ma e in which he
equilib ium posi ions o he nuclei a e epea ed pe iodically in space. The e o e, all he
p ope ies o a c ys al can be speci ied ega ding a uni cell ( he spa ial pe iodici y) and
he a oms ha i con ains. In Fig. 2.2 a c ys al la ice o med by a wo-a om uni cell is
illus a ed.
A uni cell is cha ac e ized by he se o la ice ec o s aαand by he posi ion o he
a oms inside he uni cell gi en by hei ec o s τI, usually known as he basis o he uni
cell. The e o e, any ansla ion Tnp oduced by a linea combina ion o in ege mul iples
nαo he la ice ec o s
Tn=n1a1+n2a2+n3a3(2.6)
will lea e he sys em in a ian , i.e., any unc ion o he sys em ( )(such as he equi-
lib ium posi ions o he nuclei o he elec on densi y) is in a ian unde such ansla ions
as
( ) = ( +Tn).(2.7)
39
Adol o O e o Fumega
This pe iodici y encoun e ed in c ys als allows he ep esen a ion o hose pe iodic
unc ions in e ms o Fou ie componen s a wa e ec o s k, which a e de ined in he
Fou ie ans o m o he eal space, know as he ecip ocal space.
Figu e 2.2: Schema ic o a c ys al la ice wi h wo a oms (g ey and whi e) pe uni cell.
Six uni cells a e depic ed, he la ice ec o s a1and a2, and he a omic basis ec o τ2
o he whi e a om.
I we conside a c ys al o med by a la ge numbe o uni cells Nc=Nc1×Nc2×Nc3,
whe e Ncα deno es he numbe o cells in each spa ial di ec ion α, we can simpli y he
p oblem by es ic ing o a disc e e se o Fou ie componen s pe iodic in ha c ys al
olume. This implies ha each Fou ie componen mus ul ill a se o pe iodic bounda y
condi ions ha , in he limi o la ge c ys al olume, es ic he wa e ec o s o:
kaα= 2π×in ege (2.8)
whe e he se o wa e ec o s ha ul ill his condi ion o m he ecip ocal la ice.
Thus, we can de ine he se o ecip ocal la ice ec o s bβas:
bβ·aα= 2πδβα.(2.9)
The ecip ocal la ice ec o s bβde e mine a olume Ω1BZ in he ecip ocal space
known as he i s B illouin zone.
The Hamil onian ha desc ibes he many-body sys em is in a ian unde la ice
ansla ions. Conside ing he p e ious chap e , we can conside he Kohn-Sham Hamil-
40
2 Symme y
onian o elec ons and he ha monic Hamil onian o phonons. They bo h commu e
wi h he ansla ion ope a o ˆ
Tn, hus eigens a es o he Hamil onian can be chosen o
be eigens a es o he ansla ion ope a o s and ice e sa. The e o e, he eigens a es o
he ansla ion ope a o can be used o block diagonalize he Hamil onian. This esul is
known as he Bloch heo em [37]
ˆ
Tnφ( ) = φ( +Tn) = eikTnφ( ).(2.10)
whe e kis a wa e ec o in ecip ocal space:
k=X
β
nβ
Nβ
bβ,(2.11)
whose ange can be es ic ed o he i s B illouin zone by aking nβ< Nβ. The e o e,
he eigens a es o he Hamil onian can be chosen wi h de ini e alue o kwi hin he i s
B illouin zone:
φk( ) = eik uk( ).(2.12)
These φk( )a e e med Bloch unc ions and we can see ha hey a e buil as he
p oduc o a plane wa e eik and a pe iodic unc ion uk( ). The Bloch heo em is he
eason why he me hods ha we ha e been desc ibing o sol ing he Kohn-Sham and he
la ice dynamics equa ions a e based on plane wa es. The e o e, he Hamil onians ha
lead o hese equa ions a e block-diagonal in k, and consequen ly Kohn-Sham equa ions
o independen elec ons (eq. (1.30)) and he secula equa ion o he ha monic phonon
spec um (eq. (1.56)) become independen o each wa e ec o kin he i s B illouin
zone. Then, we can compu e he eigen ec o s independen ly o each kand ob ain a se
o eigen alues labeled by εi,k.
In he mac oscopic limi o he c ys al he spacing be ween kpoin s, gi en by eq.
(2.12), ends o ze o and hence we can ake a con inuum limi o k. This esul s in he
concep o con inuum ene gy bands εi(k).
Figu e 2.3 shows a schema ic o elec onic ene gy bands o a me al and o a band
insula o . In gene al, we can de e mine he occupied le els as hose ha lay below he
Fe mi ene gy. The ene gy bands ha lay well below he Fe mi ene gy co espond o co e
elec ons. We can no ice ha he ene gy dispe sion in ko hese bands is p ac ically ze o.
This means ha co e le els a e well localized ha ba ely su e he e ec o neighbou
a oms, being ha he eason ha jus i ies he use o pseudopo en ials and a omic-like
unc ions o desc ibe hem. Nea he Fe mi le el, we ound he alence bands below
he Fe mi le el and he conduc ion bands abo e. I he e is a gap be ween he highes
ene gy alence band and he lowes ene gy conduc ion band he sys em is an insula o .
A band insula o equi es he comple e illing o he band, wi h degene a e spin s a es.
A di e en kind o insula ing beha iou , e.g., he Mo insula o , will need a symme y
b eaking d i en by elec onic in e ac ions. A me allic beha iou a ises o he gapless
case. The ene gy dispe sion o hese bands is in gene al di e en . Howe e , localized
41
Adol o O e o Fumega
o de o g asp he idea o opology. Thus, di e en signa u es o de ec he opological
cha ac e o ma e ials a e de ined o he han he Che n numbe . C ys al symme ies, like
ime e e sal o in e sion play a key ole in he Be y Cu a u e o a sys em.
In his chap e we ha e analyzed he undamen al concep o symme y in he ame
o condensed ma e physics. A de ini ion o symme y g oup has been p o ided.
Poin g oups, ansla ions and ime- e e sal symme y g oups we e desc ibed. Fi-
nally, a b ie in oduc ion o opological in a ian s was also gi en.
48
3 S a is ics and T anspo
“Why do hey called i ush hou
when no hing mo es?"
— Robin Williams
In his small chap e we will p esen some impo an quan i ies ha help o e eal he
physical beha iou o a sys em. We will p o ide a de ini ion o he densi y o s a es pe
uni ene gy and he dis ibu ion unc ions o elec ons and phonons in he mal equilib-
ium. Amongs o he hings, he analysis o he densi y o s a es allows o de e mine he
s abili y o a sys em as well as o compu e di e en equilib ium p ope ies. Mo eo e , we
will de ine anspo coe icien s h ough he Bol zmann anspo equa ion. We will see
how o compu e hem om i s p inciples calcula ions. Thus, connec ing he nanoscale
desc ip ion and he mac oscopic p ope ies o a ma e ial.
3.1 The mal Equilib ium
One o he mos impo an quan i ies ha we can ob ain when sol ing he many-body
p oblem is he densi y o s a es pe uni ene gy DOS(E):
DOS(E) = X
i
1
Ω1BZ Z1BZ
dkδ(εi,k−E)(3.1)
whe e εi,kcan ep esen he ene gy o an elec on o phonon. Fo independen pa icle
app oxima ions, like he Kohn-Sham o elec ons o he ha monic one o phonons, eq.
(3.1) co esponds o he numbe o s a es pe uni ene gy. The dimensions o a sys em,
i s symme ies and pa icle in e ac ions play a undamen al ole de e mining he shape
o he DOS a gi en con igu a ion migh p esen .
The numbe o occupied s a es pe uni olume a a gi en ene gy o a sys em in
he mal equilib ium a a empe a u e Tis gi en by he p oduc o he densi y o s a es
and a p obabili y dis ibu ion unc ion (E). Since phonons a e bosons, a Bose-Eins ein
dis ibu ion unc ion BE(E)is applied:
49
Adol o O e o Fumega
BE(E) = 1
eE−µ
kBT−1
,(3.2)
whe e kBis he Bol zmann cons an and µis he chemical po en ial. In a di e en
way, elec ons a e e mions and hence hey will be uled by a Fe mi-Di ac dis ibu ion
unc ion FD(E):
FD(E) = 1
eE−µ
kBT+ 1
,(3.3)
whe e he chemical po en ial µis called he Fe mi le el EFwhen T= 0.
Figu e 3.1: Schema ic o di e en elec onic DOS cases. The Fe mi le el is depic ed as
a dashed line.
50
3 S a is ics and T anspo
Figu e 3.1 shows a schema ic o di e en DOS si ua ions. A semiconduc o o in-
sula o is shown in he op panel. No e he p esence o an ene gy gap abo e he Fe mi
le el. In he middle panel a me allic case is exempli ied. In he lowes panel an uns able
con igu a ion is depic ed. When a high densi y o s a es occu s a he Fe mi le el o a
gi en con igu a ion, he sys em migh be uns able. This high DOS a he Fe mi le el can
be seen as a hin ha p omp s o a phase ansi ion o a lowe ene gy g ound s a e o
he sys em. A small pe u ba ion will lead he sys em o a di e en g ound s a e. In he
amewo k o a mean ield heo y, i can be shown ha he sys em ge s s able by b eaking
a symme y. An o de pa ame e is associa ed o ha ansi ion. No e ha a peak in
he Fe mi le el could occu , bu in such cases elec on co ela ions o opological ea u es
would be equi ed.
As we ha e s a ed in Chap e 1, we a e dealing wi h a many-body p oblem in which
di e en deg ees o eedom compe e. The e o e, he pa h he sys em akes o ge s a-
bilized depends on ene gy conside a ions. Fo ins ance, he sys em migh b eak ime-
e e sal symme y and become a e omagne . In he Kohn-Sham desc ip ion he spin
degene acy is b oken and one o he spin channels ge s mo e occupied han he o he .1
This is he case o a S one e omagne [55]. Al e na i ely, he sys em migh unde go
a s uc u al ansi ion by b eaking spa ial symme y. Jahn-Telle dis o ions [56] and
cha ge densi y wa es [57] a e examples o s uc u al ansi ions. Elec onic in e ac ions
migh also play a undamen al ole d i ing he sys em o an insula ing scena io. This is
known as Mo insula o beha iou [58]. In all hese phenomena, and in many o he s, a
symme y is spon aneously b oken lowe ing he DOS a he Fe mi le el, and hus lowe ing
he ene gy o he sys em. The eme gence o gaps o pseudogaps2a ound he Fe mi le el
is associa ed o ha ansi ions. Apa om ha , no e ha empe a u e is in oduced
h ough he Fe mi-Di ac dis ibu ion unc ion (eq. (3.3)), and i will play a majo ole in
he s abili y o a gi en con igu a ion. The DOS can be used o compu e di e en physical
quan i ies like he in e nal ene gy o he speci ic hea .
Finally, le us de ine he so called p ojec ed densi y o s a es (PDOSj(E)). I p o ides
he con ibu ion o a pa icula a om o o bi al j o he o al DOS. In he case o phonons
he p ojec ion is pe o med along he pola iza ion ec o o one o he modes and a oms.
Meanwhile, in he he case o elec ons he p ojec ion can be pe o med on a se o
o hono mal s a es ϕj ha can ep esen an a omic o bi al o a localized s a e o a gi en
a om.
PDOSj(E) = X
i
|hϕj|φii|2
Ω1BZ Z1BZ
dkδ(εi,k−E),(3.4)
whe e φican be seen as he Kohn-Sham s a es de e mined in he elec onic p oblem.
Thus, he o al DOS is gi en by adding all he con ibu ions o he p ojec ed PDOSj(E).
Analysis o he DOS and PDOS will be c ucial in he s udies p esen ed in he second pa .
1The DOS o each spin channel is usually ep esen ed wi h di e en sign. Posi i e o majo i y spin
and nega i e o he mino i y channel.
2Pseudogaps a e ene gy gaps ha occu a ce ain kpoin s o he B illouin zone.
51
Adol o O e o Fumega
3.2 T anspo P ope ies
In he p e ious sec ion we ha e in oduced he concep o densi y o s a es and he
dis ibu ion unc ions o phonons and elec ons. As we ha e seen, hese unc ions p o ide
he numbe o occupied s a es o a sys em in he mal equilib ium. In his pa we will
analyze he si ua ion when an elec ic ield Eo he mal g adien ∇Tis p esen in he
c ys alline solid. In such cases, a s eady-s a e low o cha ge o hea is es ablished as a
consequence o he ex e nal ields and he in e nal sca e ing p ocesses ying o es o e
equilib ium. This phenomenon is known as anspo , and he e o e a non-equilib ium
s a is ical heo y is equi ed o s udy i . We will make use o he Bol zmann anspo
equa ion o ul ill his ask.
In he p esence o ex e nal ields, he Bol zmann o mula ion conside s ha he sys em
unde goes a small de ia ion om he equilib ium dis ibu ion. Hence, a linea expansion
on ime can be pe o med on he dis ibu ion unc ion ha desc ibes he s eady-s a e
ha is es ablished. This leads o he ollowing equa ion:
d
d =∂
∂ o ce
+∂
∂ di
+∂
∂ sca
= 0,(3.5)
whe e we can see ha a s eady low is achie ed by he compe i ion be ween he
de ia ions om he equilib ium dis ibu ion caused by he ex e nal ields ( i s e m), he
di usion o he pa icles (second e m) and he sca e ing p ocesses ying o es o e he
equilib ium ( hi d e m). We will analyze sepa a ely elec onic and phonon anspo .
Di e en app oxima ions, and hence di e en le els o heo y, will be used o each kind
o pa icles. In o de o unde line ha c ys als a e he kind o sys ems ha we a e going
o s udy, om now on we will explici ly include he k-dependence in he de elopmen o
hose anspo app oxima ions.
The elaxa ion ime app oxima ion is a widely used app oach o sol e he Bol zmann
anspo equa ion o elec ons. I is based on he idea ha he sca e ing p ocesses can
be ela ed o a elaxa ion ime τ(k):
∂
∂ sca
=− (k)− 0(k)
τ(k),(3.6)
whe e he nume a o desc ibes he de ia ion o he dis ibu ion (k) om he equilib-
ium 0(k), and he elaxa ion ime τ(k)p o ides how he sys em e u ns o equilib ium.
In oducing his las equa ion on eq. (3.5), and linea izing i by conside ing ha he
ex e nal ields a e weak and hence he de ia ion o he dis ibu ion om equilib ium is
small, we a i e o he ollowing equa ion o he pe u bed dis ibu ion unc ion (k):
52
3 S a is ics and T anspo
(k) = 0(k)−τ(k) (k)∂ 0(k)
∂T ∇T+∂ 0(k)
∂E(k)eE
= 0(k)−τ(k) (k) 2
0(k)
kBTe
E(k)−µ
kBT(E(k)−µ)
T+eE,
(3.7)
whe e E(k)is he elec onic ene gy and (k)is he elec onic g oup eloci y:
(k) = 2π
~
∂E(k)
∂k.(3.8)
The e o e, eq. (3.7) p o ides he elec on popula ion pe u bed by a weak elec ic
ield and a small empe a u e g adien . No e ha in an independen elec on heo y like
he Kohn-Sham, he dis ibu ion is gene alized o each elec onic s a e wi h ene gy εi(k).
Now ha we know he pe u bed popula ion o elec ons p oduced by he ex e nal
ields, we can analyze he cu en s ha eme ge leading he sys em o he s eady s a e.
The elec on cu en densi y Jeand he hea cu en densi y JQcan be de ined as:
Je=2e
8π3Z (k) (k)dk
JQ=2
8π3Z(E(k)−µ) (k) (k)dk,
(3.9)
and in oducing eq. (3.7) we a i e a he ollowing ela ionships:
Je=e2K0E−eK1
T∇T
JQ=eK1E−K2
T∇T,
(3.10)
whe e he Kn e ms a e in eg als de ined as:
Kn=1
4π3~ZZ τ(k)(E(k)−µ)n (k) (k)
| (k)|−∂ 0(k)
∂E(k)dSdE, (3.11)
whe e he in eg al o e olume in khas been con e ed in o an in eg al o e su aces
o cons an ene gy. This is qui e con enien since only s a es nea he Fe mi le el (o
chemical po en ial) con ibu e o he anspo . Mo eo e , he elaxa ion ime is usually
app oxima ed as a cons an τ. This is known as he cons an elaxa ion ime app oxima-
ion. Apa om ha , in ab ini io calcula ions, he so-called igid-band app oxima ion
is commonly in oduced. I assumes ha he elec onic band s uc u e is independen o
changes in empe a u e o chemical po en ial.
No e ha eq. (3.10) shows ha ei he an elec ic ield o he mal g adien can gene a e
bo h an elec on o hea cu en . Such ela ionship allows us o de ine he usual anspo
53
Adol o O e o Fumega
coe icien s (elec ical conduc i i y, elec onic he mal conduc i i y, Seebeck coe icien ,
e c.) ha can be measu ed in expe imen s.3Thus, hey p o ide a clea connec ion
be ween mac oscopic p ope ies o ma e ials and hei nanoscale desc ip ion.
The iso he mal elec ical conduc i i y σp o ides he ela ion Je=σEbe ween he
applied elec ic ield and he cu en gene a ed in he ma e ial. I is gi en by:
σij =e2τ
4π~ZE=µ
i(k) j(k)
| (k)|dSE.(3.12)
Fo me als he elec onic he mal conduc i i y κecan be well app oxima ed by he
Wiedemann-F anz law h ough he elec ical conduc i i y as:
κe=(πkB)2
3e2Tσ, (3.13)
whe e he a io κe/σ is known as he Lo enz numbe . Fo semiconduc o s and insu-
la o s he e is no subs an ial cha ge low, hence hea anspo is go e ned by phonons.
We will analyze i below.
The ela ionship be ween he elec ic ield gene a ed in a ma e ial by a he mal g a-
dien E=Ss∇Tis uled by he Seebeck coe icien Ss. I is gi en by:
Ss=(πkB)2eT
3σ∂K0
∂E E=µ
.(3.14)
No e ha he Seebeck coe icien , also known as he he mopowe , is independen o
he sca e ing a e in he cons an elaxa ion ime app oxima ion. Thus, i will be one o
he mos accu a e coe icien s ha we can compu e in ha app oach. The he mopowe
will play a undamen al ole o he moelec ic applica ions.
In Pa II we will make use o he Bol zT ap2 code [59] o sol e he Bol zmann ans-
po equa ions in he cons an elaxa ion ime app oxima ion and ob ain he elec onic
anspo coe icien s.
In he s udy o phonon anspo p ope ies only he mal g adien s will be conside ed
as he ex e nal ields. The e o e, in his case, he linea ized Bol zmann anspo equa ions
lead o a pe u bed dis ibu ion unc ion
(k) = 0(k)−τ(k) (k)∂ 0(k)
∂T ∇T, (3.15)
whe e now he equilib ium dis ibu ion unc ion 0(k)co esponds o he Bose-Eins ein
dis ibu ion gi en by eq. (3.2). In he ha monic app oxima ion o phonons he pe u bed
dis ibu ion o eq. (3.15) is applied o each phonon mode.
Di e en ly om he elec onic case whe e he elaxa ion ime τ(k)was conside ed
a gi en cons an , o he case o phonons we can compu e he elaxa ion ime using he
3No e ha hese coe icien s a e second ank enso s ha p o ide he ela ionship be ween he ex e nal
ields and he cu en ha hey gene a e in he ma e ial.
54
3 S a is ics and T anspo
anha monic app oxima ion desc ibed in Chap e 1. I can be compu ed as he in e se o
he imagina y pa o he sel -ene gy (eq. (1.63)).
The la ice he mal conduc i i y κlp o ides he ela ionship JQ=−κl∇Tbe ween
a he mal g adien and a hea low. I can be compu ed as:
κij
l=1
kBT2ZE(k)τ(k) 0(k) ( 0(k) + 1) i(k) j(k)dk(3.16)
whe e usually he phonon ene gy is subs i u ed by he angula phonon equency as
E(k) = ~ω(k). Mo eo e , i is common o ob ain he o al κlas a summa ion o he
con ibu ions o each phonon mode.
In he s udies o Pa II we will compu e he la ice he mal conduc i i y using he
ShengBTE code [60].
Finally, now ha we ha e de ined he di e en anspo coe icien s, we can see how
hey a e coupled in a he moelec ic ma e ial. A he moelec ic ma e ial is cha ac e ized
by showing a la ge he moelec ic e ec [61], i.e., a he moelec ic ma e ial can gene a e
an elec ic po en ial ∆V om a empe a u e g adien ∇T(and ice e sa) as:
∆V=−Ss∇T. (3.17)
The e o e, his class o ma e ials p esen po en ial applicabili y in he a ea o e icien
ene gy managemen . As shown in eq. (3.17), a high Seebeck coe icien Ssis equi ed
in o de o ha e a good he moelec ic. Howe e , his is no he only quan i y o pay
a en ion when designing a good he moelec ic.
When a empe a u e g adien is imposed o a he moelec ic ma e ial, i s cha ge
ca ie s end o di use om he ho end o he cold end. A ne cha ge is o med a
he cold side (nega i e in he case o elec ons and posi i e o holes). This p oduces an
elec os a ic po en ial. Thus, an equilib ium is eached be ween he chemical po en ial
o di usion and he elec os a ic epulsion o he o al ne cha ge a he cold end. A
la ge he moelec ic e ec will be de eloped i he ma e ial is able o main ain he he mal
g adien , i.e., i i p esen s a low he mal conduc i i y κ, and also i i shows a high ca ie
mobili y, i.e., i i p esen s a high elec ical conduc i i y σ. The coupling be ween he
di e en anspo coe icien s ha de ines a good he moelec ic is in oduced h ough
he so-called he moelec ic igu e o me i zT as
zT =S2
sσ
κT. (3.18)
The e o e, he goal when designing a good he moelec ic is o maximize he igu e o
me i . Mo eo e , no e ha a good he moelec ic con ains only a single ype o ca ie , i
will be ei he a n- ype o a p- ype semiconduc o . This is due o he ac ha i we had
elec on and hole conduc ion a he same ime, he ne cha ge a he cold end would be
ze o and he induced elec ic ield would no a ise.
55
Adol o O e o Fumega
In his chap e we ha e in oduced he undamen al concep o densi y o s a es
and how i is used along dis ibu ion unc ions o compu e some o he p ope ies
ha desc ibe a ma e ial in equilib ium. Mo eo e , we ha e p esen ed he Bol zmann
anspo equa ion ha allows o de ine and compu e di e en anspo coe icien s.
The ing edien s o ind a good he moelec ic ma e ial we e also exposed. All his
p o ides he connec ion be ween he nano and he mac oscale o a ma e ial.
56
Pa II
S udies on C ys alline T ansi ion Me al
Compounds
57
Adol o O e o Fumega
S(T→∞)=−kB
e
∂log g
∂N (4.10)
whe e kBis he Bol zmann cons an , eis he elec on cha ge and Nis he numbe o
elec ons. In his pape he au ho s ob ain he Seebeck coe icien in di e en si ua ions.
We will show some o he esul s in o de o cla i y he equa ion ha we will use.
Fo a sys em o Nspinless e mions wi h NAsi es o be occupied, he numbe o
possible con igu a ions gis:
g=NA!
N!(NA−N)! (4.11)
using equa ion (4.10), he so-called Heikes o mula is ob ained:
S(T→∞)=−kB
elog (1 −η)
η(4.12)
whe e η=N/NA. We wan now o ind an equa ion simila o (4.12), bu o he
case o e mions wi h spin. We will be in e es ed in he case in which he epulsion ene gy
be ween e mions ( o ou pu pose hese e mions will be elec ons) is la ge han he
he mal ene gy, so in each si e he e can be jus one pa icle. The numbe o possible
con igu a ions is he one seen o spinless e mions (4.11) mul iplied by a 2N ac o ha
akes in o accoun he spin degene acy. Hence, we ge :
g=NA!
N!(NA−N)!2N(4.13)
so he Seebeck coe icien ha we ob ain is:
S(T→∞)=−kB
elog 2(1 −η)
η(4.14)
Now ha we ha e seen how o in oduce he spin degene acy, we a e in he posi ion
o expand equa ion (4.13) o accoun o he 2go bi al degene acy ha occu s a he
bo om o he conduc ion band in elec on-doped STO. We will in oduce heu is ically a
ac o 2gin eq. (4.13), ha weighs he 2gdegene acy in a simila way as he ac o 2
does o he spin degene acy:
g=NA!
N!(NA−N)!2N N
2g(4.15)
so inse ing his in equa ion (4.10) we ob ain:
S(T→∞)=−kB
elog 2 2g
(1 −η)
η(4.16)
bu we ha e no alked abou he shape o his ac o 2gye . Ou goal is o ind
an exp ession ha ep oduces he degene acy beha io in di e en s ain si ua ions. We
64
4 T anspo P ope ies o S TiO3
will model i as a unc ion o he numbe o elec ons in he dxy o bi al, Nxy and he o al
numbe o elec ons in he 2gmani old, N 2g. Figu e 4.1 shows he h ee limi ing cases
we will ha e o conside in o de o model he degene acy in e e y case. The uns ained
case implies all 2gbands a e degene a e, and hence he e is a iple o bi al degene acy.
I ensile (comp essi e) s ain occu s, he dxy (xz/yz) band/(s) lies (lie) lowe in ene gy
and hen a single (double) degene acy occu s.
Figu e 4.1: We show he h ee limi ing cases o he 2gdegene acy. In he case wi hou
s ain (middle panel), all he o bi als a e degene a e ( 2g= 3). I s ain is applied, degen-
e acy is b oken owa ds ha ing a lowe -lying single ( ensile s ain: 2g= 1, op panel) o
a lowe -lying double (comp essi e s ain: 2g= 2, bo om panel).
So we encoun e ha he ac o 2gmus be a unc ion o x=Nxy/N 2g, i s alue
mus be con ained in he in e al [1,3], and he maximum being equal o 3a x= 1/3.
Conside ing all hese ac s, we a e going o i he poin s shown in Fig. 4.1 o he ollowing
unc ion (we chose a Gaussian because i is he simples smoo h unc ion ha one can
hink o ha has only one adjus able ex emum in he [1,3] in e al):
2g=γ+α
βpπ
2
e−2((x−1/3)
β)2(4.17)
whe e he cons an s α,βand γa e comple ely de e mined by he h ee limi ing cases
shown in Fig. 4.1. A plo o he ac o as a unc ion o xcan be seen in Fig. 4.2.
In some p e ious wo ks he weigh o he degene acy in he he mopowe has been
s udied [101]. Conside ing eq. 4.16 and he ange o alues ha he degene acy ac o
2gcan ha e ([1,3]), he maximum a ia ion in S ha can be in oduced due o he
degene acy ∆Smax
(T→∞)is:
∆Smax
(T→∞)=−kB
e(log [3] −log [1]) ∼ −95 µV/K (4.18)
Howe e , in any ealis ic si ua ion, he modi ica ion will no be ha la ge.
We ha e ound an analy ical unc ion o he high empe a u e he mopowe om
a s a is ical calcula ion. A degene acy ac o o he 2gmani old was in oduced as a
65
Adol o O e o Fumega
Figu e 4.2: 2g unc ion ha esul s om eq. 4.17. This degene acy ac o ul ills he
h ee limi ing cases analyzed in Fig. 4.1 o he e ec o s ain in a 2gelec on sys em.
I accoun s o he o bi al degene acy o he sys em as a unc ion o s ain. Nxy and N 2g
need o be de e mined ab ini io.
unc ion o Nxy/N 2g. This p ocedu e will make possible o calcula e he en opy- ela ed
e m S(T→∞) om ou DFT calcula ions jus by compu ing Nxy/N 2g o each s ain
in oduced in he STO s uc u e.
Wha we ha e done so a is being able o decouple he in luence in he mopowe
a ia ions wi h s ain ha comes om elec onic degene acies and om band s uc u e
modi ica ions (such as e ec i e mass changes in a pa abolic-band desc ip ion), ha s ain
will also in oduce.
4.2.3 Resul s o he Model on S TiO3
We ha e un calcula ions in S TiO3simula ing di e en s ain si ua ions by ixing he
la ice pa ame e a o ha o he di e en s anda d pe o ski e subs a es men ioned in he
compu a ional p ocedu es sec ion. Fo each a, we ha e op imized he la ice pa ame e
c. We ha e no conside ed oxygen oc ahed al il s, since his kind o dis o ions a e
ha dly dependen on he impu i y in oduced as dopan . The e o e, we ha e limi ed
ou analysis o a cell olume dis o ion caused by biaxial s ain. We ha e ound ha
ensile (comp essi e) s ain inc eases ( educes) he uni cell olume (as shown in Fig. 4.3.
This has been expe imen ally epo ed p e iously o o he oxides [102] and also o his
sys em [103]. In Fig. 4.3 we plo he e olu ion o he uni cell olume wi h s ain. We
obse e ha , ensile s ain leads o an inc ease in uni cell olume. The olume educ ion
ha occu s o comp essi e s ain leads o a smalle Ti-Ti dis ance, which inc eases he
hopping in eg als and hence leads o an inc eased bandwid h. This bandwid h inc ease
should be ela ed o a dec ease in e ec i e mass.
66
4 T anspo P ope ies o S TiO3
Figu e 4.3: Bandwid h and uni cell olume o di e en s ain si ua ions. We see ha
bandwid h dec eases wi h olume and olume is inc eased as posi i e ( ensile) biaxial
s ain is applied.
Wi h hese s uc u es we ha e pe o med elec onic s uc u e calcula ions including
spin-o bi coupling wi h he TB-mBJ exchange-co ela ion po en ial, ha allows o gi e
an accu a e band gap wi hou addi ional compu a ional cos . Figu e 4.4 shows he band
s uc u es (only he conduc ion bands close o i s bo om) o di e en s ain si ua ions
oge he wi h he co esponding DOS’s on he same ene gy scale. GSO co esponds o
he ensile s ain limi and LAO o he comp essi e one. We can obse e ha he o me
leads, as explained abo e, o a educ ion o he 2gbandwid h, oge he wi h some shi s
in he DOS peaks. The bo om o he conduc ion band is a Γ. We can see also ha he
posi ion o he unoccupied 2gbands is highe in ene gy o comp essi e s ain and lowe
o he ensile s ain case.
Figu e 4.5 shows he ene gy bandgap a ia ion wi h cell olume and hence wi h s ain.
I can be seen ha an inc ease in he la ice pa ame e o he subs a e p oduces a dec ease
in he ene gy bandgap.
We ha e calcula ed he numbe o elec ons in he dxy o bi al, Nxy and he o al
numbe o elec ons in he 2gmani old, N 2g. Bo h alues we e ob ained by in eg a ing
inside he Ti mu in- in sphe e. We will assume ha he deg ee o localiza ion will be
iden ical o he h ee 2go bi als and hence in eg a ing inside he mu in- in sphe es is
su icien o ou pu poses since we a e only in e es ed in he a io. This a io is needed o
ob ain he degene acy ac o 2gand consequen ly he he mopowe a high empe a u e
S(T→∞)as a unc ion o doping.
67
Adol o O e o Fumega
Figu e 4.4: Band s uc u e ep esen a ion and densi y o s a es (DOS) calcula ed o
di e en s ain alues. I can be seen how he bandwid h inc eases o comp essi e s ain
and dec eases o ensile s ain.
Figu e 4.5: Ene gy bandgap a ia ion wi h cell olume. We see ha ensile s ain de-
c eases he ene gy gap.
Figu e 4.6 shows he degene acy ac o dependence wi h doping o he di e en
s ain si ua ions analyzed. We can see he e, as a unc ion o elec on doping (in e ms
68
4 T anspo P ope ies o S TiO3
o elec ons doped pe uni cell, o Nb a oms subs i u ing Ti, hus adding one ex a
elec on o he conduc ion band pe dopan ). We can see ha a low doping, he h ee
limi ing cases we discussed abo e a e ob ained (in he pu e single-ion pic u e), wi h he
uns ained case ha ing h ee old degene acy and he comp essi e o ensile s ain cases
ending o a wo old degene a e s a e o a single , espec i ely. As doping inc eases, he
2gbands become mo e hea ily popula ed and he ac ual band spli ings become less
and less impo an . Abo e 5% doping, he degene acy ac o becomes e y close o 3,
almos independen o s ain. This illus a es ha , e en hough in p inciple s ain can
une degene acies (by emo ing hem), he mopowe a high empe a u es migh no be
a ec ed i doping is subs an ial. This limi will o cou se depend on he sys em, bu o
STO we see ha i is ob ained (wi hin a igid band app oxima ion) a alues which a e on
he o de o hose sough o in he case o TE applica ions (∼1020 cm−3). The impo an
poin o no ice he e is ha he e ec o s ain on STO will no damage he TE e iciency
a high empe a u e by educing o bi al degene acies, when wo king a hose high doping
le els we a e discussing.
Figu e 4.6: Degene acy ac o as a unc ion o doping. In he low doping egime he
degene acy ac o s collapse o he h ee limi ing cases exposed in Fig. 4.1, so hey become
la gely dependen on s ain. As we inc ease he doping le el he ac o becomes s ain
independen and ends o 3as he 2gbands become mo e popula ed. Ou calcula ions
show ha his occu s be ween 2-5 % Nb-doping. In he case o uns ained STO we see
ha he degene acy ac o is comple ely independen o he doping le el since he e is
always a iple degene acy in ha case. This igu e sugges s ha degene acy e ec s anish
abo e ∼5% doping.
In oducing he esul s o he degene acy ac o 2gin eq. (4.16), we can ob ain he
he mopowe a high empe a u e S(T→∞)as a unc ion o doping o he i e s ain si u-
69
Adol o O e o Fumega
a ions ha we a e analyzing. Once S(T→∞)is compu ed, we ha e o sol e he Bol zmann
anspo equa ions o ob ain he pa ame e A. A e ha , we a e in he posi ion o ge
he alue o T0using eq. (4.4) and consequen ly de e mine he h ee pa ame e s o ou
model (eq. (4.2)). We mus wa n he eade ha he Bol zmann anspo equa ions used
o compu e he pa ame e Awould be less accu a e a lowe empe a u es (in pa icula
he phonon d ag e m is no included). Consequen ly, we will es ic ou conclusions o
analyze he e olu ion wi h s ain and no ocus oo much on he ac ual alues p edic ed
o he he mopowe . Also, disc epancies wi h expe imen al alues could be due o he
use o he cons an elaxa ion ime app oxima ion which is used ou inely bu could ha e
i s limi a ions [104].
Figu e 4.7 shows six ep esen a ions o he he mopowe ob ained using he model
ha we p opose (eq. (4.2)) a di e en doping le els o he i e s ain cases analyzed,
u ilizing he alues o he pa ame e s ha we ha e jus calcula ed. We can see ha
he high empe a u e he mopowe dec eases in absolu e alue as we inc ease he doping
le el, as one would expec . The pa ame e s A,S(T→∞),T0and also he ob ained e ec i e
masses a e summa ized in Table 4.1.
Figu e 4.8 shows he dependence ha e ec i e masses ha e wi h s ain o he high
doping egime. We can obse e he e ha he heo e ical e ec i e masses ob ained in-
c ease hei alue as doping inc eases. Howe e , eq. (4.9) p oposed o ob ain he e ec i e
masses om pa ame e A is jus a mechanism o ge he e ec i e mass endency wi h
s ain. As we ha e said, i will only be alid in he high doping egime so we will ocus
on analyzing he e olu ion o he e ec i e mass wi h s ain bu no so much on i s ac ual
doping dependence, since his will be qui e dependen on he ype o dopan . This ea u e
is analyzed in Re . [105] concluding ha he e ec i e mass will also depend on he kind o
de ec in oduced o dope STO and he kind o dis o ions hese in oduce in he la ice.
Coming back o Fig. 4.7, we can analyze now each doping egime, we see ha o low
doping le els he s ain endency is gi en by he high empe a u e Seebeck S(T→∞), i.e.
we see in Fig. 4.7, o 0.1% doping (∼1.7×1019 cm−3) ha he Seebeck e olu ion wi h
empe a u e ollows he same s ain dependence han S(T→∞). The uns ained case (wi h
aequal o ha o STO) is he one ha has he la ges Seebeck alue (in absolu e alue) a
any empe a u e. Meanwhile, a high doping le els, g ea e han o equal o 2%, acco ding
o he esul s shown in Fig. 4.6, S(T→∞)s ops being s ain dependen . Consequen ly, he
only dependence on s ain in he e olu ion o he Seebeck wi h empe a u e comes om
he Apa ame e . We can easily check ha he cu es in he 2%,4% and 50% doping
le els (∼3.3×1020,∼6.6×1020 and ∼8.4×1021 cm−3 espec i ely) ollow he s ain
dependence gi en by pa ame e Ain Table 4.1, i.e., he in e se beha iou ha e ec i e
masses ha e wi h s ain (see Fig. 4.8). This comes abou due o he ac ha 2gbecomes
e y close o 3 and s ain independen a hose doping le els. Hence, he only changes
o he he mopowe occu ia a ia ions o he ene gy- anspo e m, in which ( o high
doping egimes o 2% o mo e) he he mopowe is enhanced (in absolu e alue) by ensile
s ain.
70
4 T anspo P ope ies o S TiO3
Figu e 4.7: Rep esen a ion o he Seebeck coe icien as a unc ion o empe a u e using
eq. (4.2) o six di e en doping egimes. The pa ame e s ob ained in each case a e
compiled in Table 4.1. I can be seen ha in he low-doping egime (below 2.0%) he
en opy- ela ed e m domina es he e olu ion wi h s ain, leading o a educ ion in he
Seebeck coe icien (in magni ude) caused by any kind o s ain. Howe e , o high doping,
i is he low empe a u e (ene gy- anspo e m) he mopowe he one ha domina es
he s ain endency because S(T→∞)becomes s ain independen (see Fig. 4.6). In ha
case, we see ha ensile s ain inc eases he he mopowe (in magni ude).
71
Adol o O e o Fumega
Table 4.1: Model’s pa ame e s o he six ep esen a i e doping alues shown in Fig. 4.7.
Values o he e ec i e masses m∗/m0ob ained using eq. (4.9) a e also shown.
Doping 0.1%A(µV )S(T→∞)(µV/K)T0(K)m∗/m0
GSO 163000 -703 232 0.145
DSO 158000 -730 217 0.150
STO 159000 -737 215 0.151
LSAT 166000 -730 228 0.145
LAO 174000 -720 242 0.140
Doping 0.5%A(µV )S(T→∞)(µV/K)T0(K)m∗/m0
GSO 195000 -592 329 0.353
DSO 199000 -605 323 0.348
STO 190000 -609 312 0.368
LSAT 199000 -607 328 0.354
LAO 210000 -596 352 0.341
Doping 1.0%A(µV )S(T→∞)(µV/K)T0(K)m∗/m0
GSO 215000 -533 403 0.509
DSO 215000 -537 400 0.513
STO 219000 -545 403 0.506
LSAT 230000 -543 423 0.489
LAO 240000 -534 450 0.473
Doping 2.0%A(µV )S(T→∞)(µV/K)T0(K)m∗/m0
GSO 241000 -481 501 0.720
DSO 245000 -483 508 0.714
STO 252000 -486 519 0.700
LSAT 265000 -485 546 0.672
LAO 272000 -481 567 0.661
Doping 4.0%A(µV )S(T→∞)(µV/K)T0(K)m∗/m0
GSO 296000 -425 696 0.932
DSO 303000 -426 711 0.917
STO 311000 -427 729 0.899
LSAT 325000 -427 761 0.870
LAO 330000 -425 775 0.868
Doping 50.0%A(µV )S(T→∞)(µV/K)T0(K)m∗/m0
GSO 1490000 -154 9690 0.995
DSO 1560000 -154 10100 0.957
STO 1590000 -154 10300 0.949
LSAT 1670000 -154 10900 0.910
LAO 1750000 -154 11400 0.878
In o de o analyze hese ends in mo e de ail, we ha e ob ained he e ec i e masses
and i s s ain and doping dependence (see Table 4.1). Despi e he ac ha we show he
e ec i e masses calcula ed in six doping cases, i is ha d o analyze i s s ain dependence
away om he high doping limi . A 2% doping, we can obse e ha he e ec i e mass
is la ge o he uni cells wi h a la ge olume ( olume inc eases o ensile s ain).
Inc easing he olume leads o a educ ion in he hopping pa ame e , smalle band wid hs
and hence a la ge e ec i e mass [106]. Abo e 2% doping, such dependence wi h s ain o
he e ec i e mass is e ained. Fo lowe doping alues, he dependence wi h s ain is mo e
complex due o he pa ial in ol emen o di e en bands. The e ec i e mass inc eases
mono onously wi h doping and i eaches alues close o 1.0 m0a abou 50% doping. In
72
4 T anspo P ope ies o S TiO3
ha doping egion, a single pa abolic band pic u e makes mo e sense han a low doping,
whe e he sole concep o a single e ec i e mass whe e all he bands con ibu e in a simila
ashion is mo e di icul o iden i y.
Figu e 4.8: E ec i e mass as a unc ion o s ain in he high-doping egime. E ec i e
mass inc eases (dec eases) wi h ensile (comp essi e) s ain and ba ely changes once all
he bands become popula ed. The limi o 50% doping, e en hough beyond he alidi y
o he igid band app oxima ion is shown as he limi ing case ound in LaAlO3/STO
in e aces.
In Fig. 4.9 we can see a compa ison be ween he he mopowe calcula ed using Bolz-
T aP (ene gy- anspo e m) and he p edic ion using ou model, in which he en opy
e m is included. This plo ein o ces he idea ha low doping egimes (0.1%) he degen-
e acy has a g ea e ec on he he mopowe e olu ion wi h s ain (degene acy is di e en
o each s ain alue). Howe e , he end wi h s ain ob ained o high doping egimes
(4%) e ains ha o he ene gy- anspo e m (Bol zT aP) and is independen o he
en opy e m (degene acy is independen o s ain).
In Re . [98] Jano i e al. analyze he di ec ional e ec i e masses dependency wi h
s ain. In o de o compa e wi h hei da a, we can ob ain a mean e ec i e mass m∗
a e
pe o ming a ha monic mean o he di ec ional e ec i e masses mΓ−X(pa allel) and mΓ−X
(pe pendicula o he s ain di ec ion) gi en in his a icle and we can compa e hese
m∗
a e wi h he e ec i e masses we ha e ob ained o 2% doping. In ou calcula ions LSAT
co esponds o biaxial s ain −1.0%, STO o 0.0% and DSO o +1.0%. The esul s a e
summa ized in Table 4.2. Thei esul s, when analyzed his way, ag ee wi h ou s in he
s ain dependence o he e ec i e mass. Howe e , he ac ual alues a e la gely doping
dependen , as we ha e discussed he e.
73
Adol o O e o Fumega
Figu e 4.13: Weigh ed phase space o he high empe a u e phase a 300 K as a unc-
ion o equency. An inc ease in he weigh ed phase space is ela ed o an inc ease in
he anha monic sca e ing a e, hus dec easing he he mal conduc i i y. The a ows
indica e he peaks associa ed o he pola modes ha une he he mal conduc i i y. (a)
Compa ison be ween cubic (black) and e agonal (g een) s uc u es. (b) E olu ion wi h
he uni cell olume.
o magni ude o he STO he mal conduc i i y is gi en by he dispe sion o he acous ic
bands, since hese modes a e he main hea ca ie s, bu also uned by he pola modes.
The mean g oup eloci y associa ed o he acous ic modes a 300 K ha we ha e ob ained
is 5.24 km/s, which is in good ag eemen wi h p e ious expe imen s [II].
In Fig. 4.13 we ha e plo ed he weigh ed phase space, as de ined in [128]; i will help
us o iden i y he main sou ces o sca e ing ha une he he mal conduc i i y o STO.
The weigh ed phase space p o ides he sca e ing a e as a unc ion o equency weigh ed
by he ha monic equencies. The e o e, i is e y sensi i e o he phonon spec a and
hence can be used o iden i y he modes esponsible o he s onge sca e ing p ocesses.
80
4 T anspo P ope ies o S TiO3
Figu e 4.14: (a)Phonon lineshape o h ee di e en olumes in he Pm3mphase a 300
K. F om op panel o bo om we p esen an inc ease in he olume o he uni cell, V0
being he expe imen al uni cell olume. A he Γpoin and be ween 100 −200 cm−1, we
iden i y he iply degene a e pola mode. Inc easing he olume dec eases he equency
o he pola mode. The band dispe sion o he acous ic modes inc eases when educing
he olume. (b) Phonon mean g oup eloci y o he acous ic bands as a unc ion o he
uni cell olume.
81
Adol o O e o Fumega
I also helps o analyze how he sca e ing a e is modi ied when di e en dis o ions, ha
change he phonon spec a, a e applied o he s uc u e. In he case o cubic STO a 300
K, we can see in Fig. 4.13a (black do s) ha he weigh ed phase space has a peak a ound
250 cm−1. A his equency, his peak is associa ed o he sca e ing p ocesses o he
pola modes. So, we can conclude ha he pola modes shown in Fig. 4.11 a e he main
sou ce o sca e ing in STO. This iply degene a e pola mode was p e iously iden i ied,
using o he me hods, as one o he main sou ces o phonon-phonon sca e ing [111, 121].
Apa om ha , in a ecen wo k, his mode was also epo ed o be elec on-phonon
ac i e [129].
The subsequen discussions will be based on how he acous ic and pola modes e ol e
unde di e en dis o ions. In pa icula shi ing he pola modes o lowe equency may
be expec ed o sca e mo e s ongly acous ic phonons lowe ing he he mal conduc i i y.
Figu e 4.14a shows he lineshape o h ee di e en uni cell olumes a 300 K o a
cubic s uc u e. We can obse e ha a dec ease in he olume o he uni cell inc eases
he dispe sion o he acous ic bands. This is ansla ed in o an inc ease o he mean
g oup eloci y o he acous ic bands (Fig. 4.14b). A educ ion o he uni cell olume
also inc eases he equency o he iply degene a e pola mode due o he educ ion o
he Ti-O bond leng h. This inc ease in he equency (depic ed in Fig. 4.11) educes he
phonon sca e ing, as can be seen in Fig. 4.13b, whe e a clea educ ion o he phase space
peak occu s by educing he uni cell olume. Hence, bo h e ec s on he acous ic and
pola modes p oduce an inc ease in he he mal conduc i i y. The calcula ed alues a e:
7.6, 13.4 and 18.0 W/mK o he la ges uni cell olume o he smalles one, espec i ely.
This inc ease o he he mal conduc i i y when he olume o he uni cell is educed is
in ag eemen wi h p e ious expe imen s on o he amily o oxides ha epo an inc ease
o κwhen p essu e is applied [130].
Figu e 4.15 shows he lineshape compu ed a 300 K when a e agonal dis o ion
is applied. This dis o ion keeps he olume o he uni cell cons an and equal o he
expe imen al one, while he c/a a io was kep ixed a 1.06. We can see ha he iple
degene acy o he pola mode is b oken as explained in Fig. 4.11. Since he cla ice
pa ame e becomes la ge i will inc ease he bond leng h o he Ti wi h he apical O
a oms o he oc ahed a, educing he equency o he mode along he z-axis as shown in
Fig. 4.15. Mo eo e , due o he inc ease o he cla ice pa ame e and he dec ease o a,
he acous ic modes change hei dispe sion as compa ed o he cubic case. The calcula ed
la ice he mal conduc i i y is κxx =κyy = 10.4,κzz = 8.0W/mK. We see ha he
e agonal dis o ion b eaks he iple degene acy o he κ- enso and educes he he mal
conduc i i y a 300 K. The degene acy b eak o he pola mode is he main esponsible
o dec easing he he mal conduc i i y. In Fig. 4.13a we can see how he peak associa ed
o he pola modes is di ided in wo peaks (g een do s) when a e agonal dis o ion is ap-
plied. The di e ence ha we obse e be ween cubic and e agonal he mal conduc i i y
would be diminished aking in o accoun he olume educ ion ha occu s in he cubic
o e agonal phase ansi ion, which p oduces an inc ease on he he mal conduc i i y.
Fu he mo e, we will see in he nex subsec ion ha including he oc ahed al o a ions in
82
4 T anspo P ope ies o S TiO3
Figu e 4.15: Phonon lineshape o a e agonal dis o ion o he cubic phase a 300 K
(c/a = 1.06). The iple degene acy o he pola mode is b oken due o his dis o ion as
explained in Fig. 4.11. The e is a so ening o he pola mode along he z-axis.
he e agonal phase will inc ease he he mal conduc i i y. Thus, he di e ence ob ained
be ween bo h phases would be educed by including hese e ec s.
The low empe a u e I4/mcm phase
Below 105 K, STO unde goes a phase ansi ion om he cubic Pm3mspace g oup
o he e agonal I4/mcm [75]. This e agonal phase is cha ac e ized by he s e ching
o he uni cell and also by a TiO6oc ahed a an iphase o a ion along he c axis [131].
The s uc u e o his phase is depic ed in Fig. 4.11. We ha e op imized he la ice
pa ame e s and he a omic posi ions. We ha e hen applied di e en s uc u al dis o ions
and s udy how hey a ec he phonon dispe sion and also he he mal conduc i i y. We
will see ha , as ob ained o he high empe a u e cubic phase, he pola modes play a
undamen al ole in he hea anspo . The op imized s uc u e ha we ha e ob ained
has a0= 3.868 Å, c0= 3.935 Å, la ice pa ame e s and α0= 6.6◦oc ahed al o a ion angle
in ag eemen wi h p e ious calcula ions [124]. Bo h c0/a0 a io and α0a e o e es ima ed
by GGA compa ed o expe imen al alues [132]. Since he op imized s uc u e is a s able
s uc u e a 0 K, we can pe o m he s udy using DFT calcula ions. Small dis o ions o
his s uc u e can be applied wi hou in oducing imagina y equencies in he phonon
spec um ha would in alida e ou analysis. In his subsec ion we will s udy he e ec on
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Figu e 4.16: Resul s o he a ia ion o he uni cell olume in he I4/mcm phase.
Oc ahed al o a ion angle and c/a a io a e kep cons an . (a) Phonon band s uc u e o
h ee di e en olumes, in ed he la ges and in blue he smalles , V0is he op imized
uni cell olume. An inc ease in he olume educes he equency o he pola modes
and he phonon dispe sion o he acous ic modes inc eases. (b) The mal conduc i i y as
a unc ion o empe a u e. An inc ease in he olume educes he he mal conduc i i y.
he spec um and he he mal conduc i i y o uni cell olume, c/a a io and oc ahed al
o a ion angle. We will keep wo o hese pa ame e s cons an while a ying he hi d one
o y o decouple hei e ec s. No e ha we will plo he he mal conduc i i y o his
low empe a u e phase o empe a u es abo e 105 K, he empe a u e o he s uc u al
phase ansi ion. We do his o be e isualize he a ia ions in κwi h each dis o ion.
Mo eo e , below 80 K he he mal conduc i i y is no plo ed since i is domina ed by
he sample dependen bounda y sca e ing. The e o e, he ealis ic empe a u e ange o
he compu ed κis be ween 80 and 105 K.
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4 T anspo P ope ies o S TiO3
Figu e 4.17: Mean g oup eloci y o he acous ic modes o he low empe a u e phase.
I s inc ease p oduces an inc ease o he he mal conduc i i y. (a) E olu ion wi h he uni
cell olume, (b) c/a a io and (c) oc ahed al o a ion angle.
Figu e 4.16a shows he phonon dispe sion o h ee di e en olumes o he uni cell.
We can see ha a educ ion o he olume inc eases he dispe sion o he acous ic bands,
hus inc easing he g oup eloci y o he compound (Fig. 4.17a). We can also see ha he
pola modes inc ease hei equency when he olume is educed. This causes a educ ion
o he phonon-phonon in e ac ion. Figu e 4.18a shows how he peak o he weigh ed
phase space associa ed o he pola modes dec eases when he olume is educed. These
wo consequences in he phonon spec um ac in a posi i e way inc easing he he mal
conduc i i y when educing he olume (see Fig. 4.16b). This end is in ag eemen wi h
he esul ob ained o he high empe a u e cubic phase. The he mal conduc i i y a
100 K o he op imized s uc u e is κzz = 16.1W/mK and κxx =κyy = 14.2W/mK.
Conside ing ha V0is o e es ima ed by GGA, hese a e in p e y good ag eemen wi h he
expe imen al alue [II] om he Lis o Publica ions and also wi h p e ious calcula ions
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Figu e 4.18: Weigh ed phase space o he low empe a u e phase a 80 K as a unc ion
o equency. The high peak a ound 200 cm−1is associa ed o he pola modes. I s
inc ease p oduces an inc ease o he anha monic sca e ing a e and hence a educ ion o
he he mal conduc i i y (a) e olu ion wi h he uni cell olume, (b) c/a a io and (c)
oc ahed al o a ion angle.
[111]. We see ha a dec ease in 1% in he uni cell olume p oduces an inc ease o a ound
30% on he he mal conduc i i y.
Figu e 4.19a shows he phonon dispe sion o h ee di e en c/a a ios. We can see
ha a educ ion o he c/a a io inc eases he dispe sion o he acous ic bands, hus
inc easing he g oup eloci y o he compound (Fig. 4.17b). We can also see ha he
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4 T anspo P ope ies o S TiO3
Figu e 4.19: Resul s o a ia ion o he c/a a io in he I4/mcm phase. Uni cell olume
and oc ahed al o a ion angle a e kep cons an . (a) Phonon band s uc u e o h ee
di e en c/a a ios, in ed he la ges and in blue he smalles , c0/a0is he op imized
c/a a io. The acous ic bands unde go a so ening, mos ly a he X poin , when he c/a
a io is inc eased. A he Γpoin , we see ha an inc ease o he c/a a io inc eases he
gap be ween he xy and z-axis pola modes. (b) The mal conduc i i y as a unc ion o
empe a u e. Inc easing he c/a a io lowe s he he mal conduc i i y.
z-axis pola mode so ens when he c/a a io is inc eased. Mo eo e , he ene gy gap
be ween his pola mode and he o he doubly degene a e pola mode inc eases wi h c/a
(see Fig. 4.11). Figu e 4.19b shows he inc ease o he he mal conduc i i y when c/a is
educed. We obse e ha a dec ease o 1% in he c/a a io p oduces an inc ease o a ound
14% on he he mal conduc i i y. The e ec o he c/a a io on he he mal conduc i i y
is ela i ely smalle compa ed o he e ec o he uni cell olume. The eason o his is
ha one pola mode so ens and he doubly degene a e ones aise hei equency in he
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Figu e 4.20: Resul s o he a ia ion o he oc ahed al o a ion angle in he I4/mcm
phase. Uni cell olume and c/a a io a e kep cons an . (a) Phonon band s uc u e o
h ee di e en angles, in ed he la ges and in blue he smalles , α0is he op imized
o a ion angle. An inc ease in he angle inc eases he equency o he pola modes.
(b) The mal conduc i i y as a unc ion o empe a u e. Inc easing he angle aises he
he mal conduc i i y.
case o he c/a dis o ion, while he h ee pola modes so en when modi ying he olume
(see Fig. 4.11), inc easing he phonon-phonon sca e ing and ha ing a la ge e ec on κ.
Figu e 4.20a shows he phonon dispe sion o h ee di e en oc ahed al o a ion angles
α. We can see ha inc easing αhas no a big e ec on he dispe sion o he acous ic
bands, compa ed o he a o emen ioned olume dis o ions o a ying he c/a a io. Mean
g oup eloci ies o he acous ic modes inc ease wi h he angle, bu ba ely compa ed o he
o he dis o ions. We can also see ha he z-axis pola mode so ens when αis educed
(see Fig. 4.11). Figu e 4.20b shows he inc ease o he he mal conduc i i y when αis
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4 T anspo P ope ies o S TiO3
inc eased. This can be unde s ood in e ms o he e olu ion o he weigh ed phase space
wi h he oc ahed al angle. I s inc ease causes a educ ion o he peak. We obse e ha an
inc ease o 1◦in αp oduces an inc ease o a ound 14% on κzz while only 3% on κxx =κyy.
The e ec o a ying αon he he mal conduc i i y is lowe compa ed o he e ec o
he wo o he dis o ions we ha e conside ed and analyzed. The main eason o his is
ha he acous ic bands a e ba ely modi ied by α, since he e is no change in he la ice
pa ame e s.
The compe i ion be ween he an i e odis o i e (oc ahed al o a ions) and e oelec-
ic phases was al eady s udied in li e a u e, by supp essing he o a ions one can in
p inciple achie e a e oelec ic phase [133–135]. This could be achie ed ia s ain en-
ginee ing since di e en oc ahed al o a ion pa e ns and angles can be ob ained [136].
We ha e shown he e he coupling be ween he oc ahed al o a ions and he pola mode
esponsible o e olec ici y and also he e ec ha his has on he he mal conduc i i y
o STO.
4.4 Conclusions
In his chap e we ha e desc ibed wo independen wo ks on anspo p ope ies o
S TiO3. They a e p esen ed oge he due o he applicabili y o STO as a he moelec ic
ma e ial.
In he i s wo k we ha e analyzed he he mopowe o STO and i s s ain depen-
dence o di e en n-doping le els, being able o decouple he di e en o igins o hese
dependencies: hose coming om elec onic degene acies and en opy- ela ed e ms om
hose coming om he ac ual modi ica ions o he band s uc u e. We ha e p oposed a
simpli ied model o he Seebeck coe icien and applied i o analyze he he mopowe
s ain dependency o STO. We ha e ob ained ha in he low-doping egime he elec on
degene acy domina es he Seebeck so any kind o s ain dec eases he he mopowe , while
a high doping he cell olume e ec domina es, i.e., inc easing olume ( ensile s ain)
leads o an inc ease in he Seebeck.
In he second wo k we ha e analyzed he he mal conduc i i y o STO. We ha e s ud-
ied he main sou ces o he mal anspo and how hese a e modi ied when a dis o ion
is in oduced in he s uc u e. We ha e analyzed bo h he high empe a u e cubic phase
and he low empe a u e e agonal one. We ha e iden i ied he acous ic bands and he
low-ene gy iply degene a e pola mode as he main hea ca ie s. Reduc ion o he uni
cell olume o he c/a a io (i.e., app oaching a cubic s uc u e) a e ound o inc ease
he la ice he mal conduc i i y. The oc ahed al o a ion angle was ound o be s ongly
coupled o he phonon-ac i e pola modes. We epo ha an inc ease o his angle aises
he he mal conduc i i y.
Conside ing oge he bo h wo ks, we can igu e ou some conclusions abou he he -
moelec ic esponse o his ma e ial. We ha e seen ha in o de o op imize his esponse
we need o inc ease he he mopowe and educe he la ice he mal conduc i i y as much
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cha ge, supposing hey become ully ionized, as is ound in o he oxides [136]. Finally,
E0_ ac is he ene gy o he con igu a ion wi hou acancies 0_ ac, his e m ep esen s
he ze o o ene gies when ob aining E om eq. (5.1). We ha e assumed ha all he
acancies o m O2and we ha e also neglec ed he elec os a ic in e ac ion be ween he
homogeneous backg ound cha ge and he cha ged acancies. This assump ion was made
conside ing p e ious wo ks [165–167] and he ac ha he mos s able compound o he
oxygen elemen is he O2molecule.
All e ms in eq. (5.1) we e calcula ed using he GGA scheme, bu Eewas compu ed
using he LDA+Uscheme o a ious U alues, since GGA leads o me allic solu ions
and he LDA+Ume hod allows o compa e he band alignmen be ween wo ealis ic
insula ing solu ions, bo h o he s oichiome ic and non-s oichiome ic sys ems.
The esul s o he ene gy o each oxygen- acancy con igu a ion using eq. (5.1) a e
shown in Fig. 5.3. We obse e ha only he con igu a ion 2_ ac_D is mo e s able han
he one wi hou acancies. This co esponds o chains o oxygen acancies pe pendicula
o he (001) and con ained in he plane o acancies (see Fig. 5.2).
Figu e 5.3: Ene ge ics o he di e en oxygen- acancy con igu a ions compu ed using eq.
(5.1). We can see ha he mos s able con igu a ion is he 2_ ac_D. This con igu a ion
co esponds o acancy chains pe pendicula o he (001) di ec ion and con ained in he
acancy plane. Only esul s wi h U= 7 eV a e p esen ed in his igu e.
Be o e analyzing in de ail con igu a ion 2_ ac_D, which is he g ound s a e, we can
explo e he o he con igu a ions. Fo he case wi hou acancies, he sys em is a non-
magne ic insula o . We also see ha his con igu a ion is qui e s able compa ed o he
o he s ha p esen oxygen acancies (2_ ac_D is he only con igu a ion wi h acancies
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5 Fe omagne ic and Insula ing Beha io o LaCoO3
ha is mo e s able han he s oichiome ic solu ion). I we pu an eye now on he wo
con igu a ions wi h one acancy, we will ealize ha one o hem is much mo e s able han
he o he . The di e ence in ene gy be ween hem is a sizable 0.37 eV/Co. This esul
sugges s us ha in he case o adding a second acancy, i could be ene ge ically a o able
o include i in an equi alen posi ion o con igu a ion 1_ ac_B, i.e, o ming chains
pe pendicula o he (001) di ec ion. We ha e demons a ed his s a emen calcula ing
he ene gy o he 7 possible 2- acancy con igu a ions. As we ha e said 2_ ac_D gi es
a s able solu ion. All he o he n= 2 cases a e also FMI solu ions, bu highe in ene gy,
excep 2_ ac_C, which is a e omagne ic me al. We ha e ied se e al alues o Uin a
wide ange, bu an ene gy gap was no opened o ha pa icula con igu a ion.
Table 5.1: Ene gies (in eV ) o he di e en con igu a ions a di e en alues o U. The
ze o o ene gies is ep esen ed by he con igu a ion wi hou acancies. I can be obse ed
ha con igu a ion 2_ ac_D is a obus g ound s a e in a wide ange o alues o U. I is
he only con igu a ion ha is s able o U≥5.5eV .
U(eV ) 4.1 5.5 6.8 8.2
1_ ac_A 5.78 5.40 5.08 5.16
1_ ac_B 1.33 0.96 0.66 0.70
2_ ac_A 2.29 1.44 0.72 0.87
2_ ac_B 2.61 1.60 1.23 1.43
2_ ac_C 1.00 -0.16 0.08 0.47
2_ ac_D 1.93 -0.24 -0.64 -0.41
2_ ac_E 3.13 2.41 2.45 2.90
2_ ac_F 1.38 1.10 0.99 1.56
2_ ac_G 2.61 1.62 1.33 1.75
Table 5.1 shows a summa y o all he ene ge ics o he con igu a ions conside ed as a
unc ion o U o ou selec ed alues o U. We can see ha he only s able solu ion ha
includes O- acancies is he one we ha e analyzed ho oughly in he pape : con igu a ion
2_ ac_D. We see ha i is he g ound s a e o a wide ange o U alues, U≥5.5eV .
The ypical U alues u ilized o his kind o cobal a es in simila calcula ions a e in he
ange o 6−8eV [159, 168, 169]. Thus, ou calcula ions show a obus end indica ing
ha con igu a ion 2_ ac_D is indeed he g ound s a e o he sys em.
Figu e 5.4 shows ha he e is a ange o U alues whe e 2_ ac_D becomes s able,
and such Uis wi hin ha ange. We ha e also seen ha con igu a ion 2_ ac_D is a
obus g ound s a e in a wide ange o U alues abo e 5.5eV . We will de e mine he
o igin o his s abiliza ion in he nex subsec ion as a change in he elec onic s uc u e
p oduced by he elec onic co ela ions.
I we analyze in mo e de ail he s uc u e o he 2_ ac_D con igu a ion ha we
ha e ob ained om ou calcula ions, we ind he ollowing cha ac e is ics ha can be
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Adol o O e o Fumega
Figu e 5.4: Ene gy o he 2_ ac_D con igu a ion as a unc ion o Ucompu ed using eq.
(5.1). The ze o o ene gies would be he s oichiome ic solu ion wi hou acancies. We
obse e ha o alues o U > 5.5eV he con igu a ion labeled 2_ ac_D becomes s able.
compa ed wi h he expe imen al esul s epo ed in Re . [156]. The la ice pa ame e
along he (001) di ec ion unde goes a con ac ion compa ed o he bulk alue o LCO.
We ha e ob ained c= 3.810 Å, which is in easonable ag eemen wi h he expe imen al
alue c= 3.767 Å[156]. The dis ance be ween LaO planes pa allel o he plane o oxygen
acancies depends on whe he one conside s he dis ance be ween wo LaO planes ha
con ain he plane o acancies o wo s oichiome ic planes. In he o me case, wi h he
plane o oxygen acancies inside, we ound ha he dis ance is 4.02 Å. In he o he case
we ound i is 3.85 Å. Expe imen ally, he same end is obse ed bu wi h a la ge a io
[160]. The e o e, we ha e ound in his subsec ion a con igu a ion o oxygen acancies ha
ag ees wi h he expe imen al mic oscopy da a ob ained o LCO ilms g own on STO.
5.3.2 Elec onic S uc u e and Magne ic beha io
We analyze now he elec onic s uc u e o con igu a ion 2_ ac_D. As we ha e al-
eady said, his p esen s a e omagne ic-insula o beha io and is ound o be a g ound
s a e. We will y i s o explain he simple ionic model ha accoun s o he magne ic
momen s obse ed and i s consis ency wi h he ab ini io calcula ions o he pa ial densi y
o s a es (DOS) o he Co a oms shown in Fig. 5.5. In i , we can see he 3 ypes o Co
a oms ha appea in he con e ged solu ion: a non-magne ic Co a om six- old coo di-
na ed and wo magne ic Co a oms wi h 5- old oxygen coo dina ion, one wi h a highe
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5 Fe omagne ic and Insula ing Beha io o LaCoO3
alue o he magne ic momen han he o he .
Figu e 5.5: Pa ial DOS o Co2+ HS, Co2+ LS and Co3+ LS compu ed in he LDA+U
scheme, o U= 7 eV . The Fe mi le el is se o he ze o o ene gies. Uppe panel, Co2+
HS: he 2gand he egmajo i y-spin s a es a e occupied, and also he dxz, dyz mino i y
o bi als. Middle panel, Co2+ LS: he 2gmajo i y and mino i y o bi als a e occupied and
so is he dz2majo i y. Lowe panel, Co3+ LS: he 2gmajo i y and mino i y o bi als a e
occupied.
F om an ionic pic u e, i we conside he uni cell shown in Fig. 5.1, we ha e La3+,
O2−and Co3+ o he s oichiome ic compound wi hou acancies. The 2go bi als o
Co3+, which a e in an oc ahed al en i onmen , a e occupied in he low-spin (LS) s a e,
see Fig. 5.6 panel c), which gi es ise o a ze o magne ic momen and an insula ing
s a e caused by c ys al- ield spli ing be ween he Co 2gand egle els ha leads o a gap
opening a ound he Fe mi le el in he simple diamagne ic con igu a ion.
I we emo e now wo oxygens om he uni cell (Fig. 5.1), we ha e he s oichiome y
La12Co12O34. S uc u ally, he e a e ou Co a oms ha a e i e- old coo dina ed and
he emaining eigh Co ca ions a e six- old oc ahed ally coo dina ed and emain non-
magne ic, sugges ing hey e ain he o iginal 3+ alence. A simple elec on coun would
nai ely imply ha he o he 4 Co a oms a e Co2+ ca ions. These a e in a i e-oxygen
en i onmen , ha could be analyzed in an ionic pic u e as an oc ahed al en i onmen
dis o ed along he z-axis. Such dis o ion b eaks he degene acy o he 2gand he eg
le els as ske ched in Fig. 5.6. The coo dina e sys em ha we use o analyze he elec onic
s uc u e is local o each Co a om wi h z lying along he O-Co- acancy di ec ion. Ou
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Adol o O e o Fumega
Figu e 5.6: Ionic model o Co2+ HS, Co2+ LS and Co3+ LS. a) Co2+ HS: he 2gand he
egmajo i y s a es a e occupied, and also he dxz, dyz mino i y o bi als. b) Co2+ LS: he
2gmajo i y and mino i y o bi als a e occupied and also he dz2majo i y. c) Co3+ LS:
he 2gmajo i y and mino i y o bi als a e occupied.
calcula ions show ha he magne ic momen alues a e consis en wi h an ionic pic u e
whe e hal o hese Co2+ ca ions a e in he LS s a e (in he 2_ ac_D panel o Fig. 5.2 he
Co a oms on he le side), and he o he hal in he HS s a e ( he ones on he igh side).
The ac ual alues o he magne ic momen s inside he mu in- in sphe es ob ained om
ou calcula ions a e app oxima ely 0.8 and 2.4 µB, which accoun o he ionic alues plus
subs an ial hyb idiza ions. Taking in o accoun Fig. 5.6: o he case o Co2+ LS (Fig.
5.6 b)), we obse e ha he 2gbands a e ully occupied and he e is one ex a elec on
occupying he lowe -lying egs a e, which is o dz2pa en age due o he missing apical
oxygen. Thus, he LS Co2+ ca ions a e in an S= 1/2s a e. In he case o Co2+ HS (Fig.
5.6 a)), he majo i y channel is ully occupied, i.e., he 2gand he egmajo i y s a es a e
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5 Fe omagne ic and Insula ing Beha io o LaCoO3
occupied, he e a e wo mino i y 2gelec ons occupying he lowe -lying double xz/yz,
which is spli om he highe -lying xy o bi al due o he missing apical oxygen in he i e-
old-coo dina ed en i onmen . This leads o S= 3/2HS s a e. This simple ionic pic u e
can be obse ed o be app oxima ely ep oduced (wi h subs an ial hyb idiza ions, as is
common in ansi ion me al oxides) in he pa ial DOS o each inequi alen Co ca ion
shown in Fig. 5.5. In his igu e we show ha he DFT calcula ions can be unde s ood
wi h he simplis ic ionic model ske ched in Fig. 5.6. The labels in Fig. 5.5 help o see he
co espondence wi h he ionic model. Spec oscopy da a con i m ha mos o he sample
is Co3+ non-magne ic and ha some magne ic Co2+ exis [156]. The da a do no disca d
he exis ence o Co3+ HS ye hey canno con i m i . Ou calcula ions show ha Co3+ HS
does no appea . No analysis o he possible exis ence o Co2+ LS ha ou calcula ions
p edic is p o ided in Re . [156], hus ou calcula ions a e in p inciple consis en wi h
spec oscopic e idences.
Figu e 5.7: Pa ial DOS Co2+ LS o wo di e en alues o U,4and 7eV espec i ely.
Uppe panel: we obse e ha he dz2o bi al (highligh ed in yellow) is no ully occupied
which esul s in a solu ion ha is no he g ound s a e. Lowe panel: we obse e in his
case ha he dz2o bi al is ully occupied, which s abilizes he 2_ ac_D con igu a ion o
become he g ound s a e.
The o al spin magne ic momen in his solu ion is 0.67 µB/Co. I we now include
spin-o bi coupling in he calcula ions, he o al o de ed momen ob ained as he sum o
lz+2szis abou 0.8µB/Co, since he magne ic Co ca ions p esen a non-negligible o bi al
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Adol o O e o Fumega
angula momen um ha becomes pa ially unquenched. This is in easonable ag eemen
wi h he sa u a ion magne iza ion o he LCO hin ilms ob ained expe imen ally o abou
0.85 µB/Co [156]. We also obse e in Fig. 5.5 ha he ene gy gap is be ween he dz2
o bi al o he Co2+ LS ( op o he alence band) and he ego he Co3+ LS (bo om o
he conduc ion band). O cou se he pa icula alue o he gap will depend on he alue
o Uchosen o he LDA+Ucalcula ions. In pa icula , o he DOS p esen ed in Fig.
5.5, he alue used was 7eV .
In he p e ious subsec ion we s a ed ha o alues o Uless han 5.5eV he 2_ ac_D
con igu a ion was no s able. This can be analyzed h ough he change in he elec onic
s uc u e ha occu s when compa ing he low-Uand high-Usolu ions. Plo s o he pa -
ial DOS o Co2+ LS o wo di e en alues o U, one o an uns able case and ano he
one o he s able case a e shown in Fig. 5.7. We can see ha he dz2o bi al o he
Co2+ LS is no ully occupied o he uns able case, while i becomes o ally occupied o
he s able case a la ge U alues. This change in he elec onic s uc u e is co ela ed
wi h he s abiliza ion o he 2_ ac_D con igu a ion. The e o e, acco ding o ou esul s,
he insula ing p ope ies o hese ilms occu na u ally due o he addi ional band spli -
ings in oduced by he non-oc ahed al en i onmen s and he co ela ion ene gy o he d
elec ons.
Table 5.2: Ene gy o he whole uni cell (12 Co a oms) o he di e en magne ic con ig-
u a ions. They a e all e e ed o EF M .
Ene gy (eV)
EFM 0.0
EAFM12.1
EAFM20.5
EAFM32.2
EAFM41.9
EAFM52.2
Finally, le us analyze he magne ic o de ha occu s in LCO. In o de o do so,
we ha e pe o med calcula ions on di e en magne ic con igu a ions in he g ound s a e
s uc u al con igu a ion 2_ ac_D. These con igu a ions a e shown in Fig. 5.8. The co -
esponding magne ic exchange coupling cons an s we ha e in oduced in o a Heisenbe g
model o he sys em a e depic ed as he a ious J’s. The ene gies o each magne ic con-
igu a ion a e shown in Table 5.2. They a e all e e ed o he ene gy o he e omagne ic
(FM) con igu a ion EF M , which is aken as ze o. To al ene gy is gi en o he whole uni
cell (12 Co a oms in o al). The a angemen o he Co2+ HS and LS is p ese ed o all
he magne ic con igu a ions and he geome y o he s uc u e is no changed. We see
ha he FM con igu a ion is he mos s able.
102
5 Fe omagne ic and Insula ing Beha io o LaCoO3
Figu e 5.8: Fe omagne ic and an i e omagne ic con igu a ions o he 2_ ac_D in he
plane o O- acancies. The spin di ec ion o each Co is depic ed wi h a ows. The exchange
in e ac ions a e labeled wi h J1, J2, J3and J4. The numbe s inside he ed ci cles ep esen
he modulus o he magne iza ion o each Co a om (1 and 3 µB) in he co esponding
ionic limi .
Mo eo e , we can ob ain he alue o he di e en exchange in e ac ions as a unc ion
o he ene gies o he magne ic con igu a ions:
J1=EAFM2+EAFM3−EAFM1−EF M
24 = 300K
J2=EAFM1+EAFM3+ 2(EAFM4−EAF M5)−EAFM2−EFM
8= 4600K
J3=EAFM1+EAFM3−2(EAFM4−EAF M5)−EAFM2−EFM
72 = 730K
J4=EAFM1+EAFM2−EAFM3−EF M
24 = 150K
(5.2)
We obse e om eqs. (8.1) ha all he he exchange in e ac ions (J’s) a e posi i e,
which en ails a e omagne ic o de acco ding o ou sign con en ion.
103
Adol o O e o Fumega
We ha e cha ac e ized in his sec ion he elec onic s uc u e o he 2_ ac_D con ig-
u a ion. We ha e ela ed he calcula ed DOS o a simple ionic model ha explains bo h
he e omagne ic o de ing wi h a consis en alue o he o al magne ic momen and he
insula ing beha io .
5.4 Conclusions
In his chap e we ha e analyzed he e omagne ic-insula ing beha io o LCO when
i is g own on (001) STO. We ha e used DFT calcula ions o analyze he elec onic
s uc u e p ope ies o LCO unde ensile s ain and o a ious s oichiome ies including
di e en oxygen acancy con igu a ions and concen a ions.
We ha e ound ha he g ound s a e o LCO when i is g own on STO is gi en
by an o -s oichiome y o he o m LaCoO2.83 p oduced by abou 6% oxygen acancies.
We ha e shown ha he acancies o m chains pe pendicula o he (001) di ec ion and
a e con ained in he plane pe pendicula o he ilm/subs a e in e ace, consis en wi h
expe imen al indings. O he s uc u al ea u es like he dis ance be ween La laye s o
he la ice pa ame e in he (001) di ec ion ag ee wi h he expe imen al measu emen s.
We ound ha he Co a oms ha lie in he plane o acancies a e a mix u e o Co2+
LS and Co2+ HS. The o al magne ic momen is in close ag eemen wi h expe imen al
measu emen s. The e omagne ic insula ing beha io o he ilm is eadily ob ained as
a g ound s a e, he gap opening occu ing na u ally in ha elec on coun due o he
appea ing c ys al ield spli ings oge he wi h he addi ion o a easonable U alue. Ou
ab ini io o al ene gy calcula ions con i m ha all he nea es -neighbo magne ic exchange
couplings a e indeed e omagne ic.
In conclusion, we can say ha he s ain in oduced by STO on LCO a o s he
p esence o O- acancy chains pe pendicula o he (001) di ec ion. Fe omagne ism a ises
om he inclusion o hose acancy chains and can be explained wi h an ionic model.
The esul s p esen ed in his wo k could lead o a be e unde s anding and he e en ual
design o o he e omagne ic-insula o oxides in which oxygen acancies could play an
impo an ole.
104
6 S -doping on In ini e-laye Nickela es
Jus a yea ago be o e he pandemic, S -doped NdNiO2was ound o be a high
empe a u e supe conduc o . This so-pu sued esul has opened he possibili y o
s udy nickela es as cup a es analogs. In his chap e we s udy he e ec o S -doping
in he elec onic and magne ic p ope ies o in ini e-laye nickela es as well as he
na u e o he holes. Ou esul s show ha doping induces a cup a e-like cha ac e
on in ini e-laye nickela es. The wo k p esen ed he e can be ound in Re . [V] om
he Lis o Publica ions and i is pa o a collabo a ion wi h An ía Bo ana’s g oup.
6.1 In oduc ion
One o he majo achie emen s in he ield o oxides was he disco e y o high-
empe a u e supe conduc o s (HTS) in 1986 [170]. Since hen, many di e en s a egies
ha e been aken o y o deciphe he o igin o HTS [171]. Cup a e nanos uc u ing in
which CuO2laye s a e a anged wi h di e en dopan s in mul iple ways has been p obably
he mos s udied ou e. Howe e , a mo e illumina ing app oach could be o subs i u e
Cu2+ wi h isoelec onic Ni1+: d9[172]. This nickel oxida ion s a e is epo ed o occu in
in ini e laye ed nickela es o he o m RNiO2(whe e R= La, Nd) [173–177]. I ook mo e
han 30 yea s o expe imen ally achie e a doped in ini e-laye nickela e [178]. In 2019, S -
doped NdNiO2was epo ed o be a supe conduc o wi h Tc∼15 K and a dome-shaped
doping dependence as occu s o cup a es [179]. In his chap e we will analyze om
i s p inciples how he elec onic s uc u e and magne ism o in ini e-laye nickela es is
a ec ed by S doping.
The pa en phase o 112 nickela es (a d9 illing), i.e. he s oichiome ic phase, is qui e
di e en om ha o cup a es. Expe imen al measu emen s show ha RNiO2 amily
is me allic and no an i e omagne ic o de is ound [173–176]. Apa om ha , and
unlike cup a es, p e ious elec onic-s uc u e calcula ions on RNiO2compounds epo
he p esence o low-lying R-5d s a es c ossing he Fe mi le el. Speci ically, he Ni 3dz2
and R 5dz2hyb idiza ion c ea es a small sphe ical elec on pocke cen e ed a he Γpoin ,
and he Ni 3dz2and R dxy hyb idiza ion c ea es ano he elec on pocke a he co ne s o
he B illouin zone. Elec ons occupying hese pocke s o igina e om he o he wise illed
105
Adol o O e o Fumega
Figu e 6.5: E olu ion o he o bi al esol ed DOS o Ni-dz2and Ni-dx2−y2s a es upon
inc easing S -doping in LaNiO2, ob ained using he AMF scheme wi h U= 5 eV.
weigh o he Ni dz2band a ound he Fe mi le el as S -doping is in oduced, lea ing a
dominan dx2−y2con ibu ion ha makes S -doped 112 nickela es a mo e cup a e-like,
single-band sys em (see Fig. 6.5). All o he abo e desc ibed ends imply ha , as S
dopan s a e in oduced in RNiO2ma e ials, some o hei elec onic-s uc u e ea u es
become close o hose o he cup a es: low-spin dopan s a es, educed cha ge- ans e
ene gy, and a single Ni dx2−y2band a ound he Fe mi le el.
Finally, he 4 ×4 supe cells wi h an a e age d8.75 illing gi en by a 25% S subs i u ion
ha allow, ia clus e ing o all he S dopan s, o one Ni in he cell o be nominally d8
( he Ni a om comple ely su ounded by i s -neighbo S ca ions in Fig. 6.2). In his
scena io, we ind ha a LS (S= 0) s a e is p e e ed in bo h RNiO2(R= La, Nd), o
he Ni ion su ounded by S a oms e en wi hin FLL. In he AMF scheme he low-spin
solu ion is no only he lowes in ene gy, bu a high-spin solu ion does no e en exis , as
a emp s o s a he sel -consis ency p ocedu e wi h a S=1 s a e o Ni2+ ion lead o a
anishing magne ic momen . The es o he Ni a oms p ese e he expec ed magne ic
momen s. Figu e 6.6 con as s he o bi al esol ed densi y o s a es o he nominally d8
Ni ca ion ( ha is su ounded by S a oms) in bo h a LS s a e and a HS s a e in his
4×4 supe cell wi hin FLL. We choose once again o show calcula ions o S -doped
LaNiO2bu he si ua ion is iden ical in he Nd-ma e ial. In he low-spin s a e he 6
2g
d2
z2con igu a ion is clea wi h he wo dx2−y2o bi als emaining unoccupied o bo h spin
channels. In he high-spin case, he 2go bi als a e also comple ely occupied, bu now
one elec on occupies he majo i y spin dz2and dx2−y2o bi als. The LS s a e o he
Ni2+ is s ongly connec ed o a educ ion in he dz2cha ac e a ound he Fe mi le el.
This can be seen in Fig. 6.6 whe e he Ni dz2band c ossing he Fe mi le el o he HS
s a e becomes ully occupied, well below he Fe mi le el o he LS s a e. This e ec is
concomi an wi h he educ ion o he La-d sel -doping e ec upon inc easing S -doping
112
6 S -doping on In ini e-laye Nickela es
desc ibed abo e. O e all, he s able LS s a e solu ion we ind gi es hen ise o an explici
cup a e-like scena io wi h planes o S=1/2 ions ha a e ligh ly doped wi h mobile low-
spin S=0 ions, a con igu a ion ha is di ec ly analogous o he low-spin S=0 Cu3+ ion
si ua ion, media ed by O-p holes as explained abo e.
Figu e 6.6: DOS o he nominally Ni2+ dopan in he 4×4 supe cell wi h 25% S -doping
in he wo di e en spin s a es s udied. HS (LS) con igu a ion is shown in he le ( igh )
panel. I can be no iced ha he lowe -ene gy LS con igu a ion leads o a deple ion o Ni
dz2s a es a ound he Fe mi le el.
We no e ha in o he in ensi ely s udied nickela es (such as in La2NiO4) he high-spin
(S=1) con igu a ion o Ni2+ is a o ed [199, 200] and no he low-spin (S=0) con igu a ion
as we ind he e. La2NiO4is s uc u ally di e en o RNiO2as i p ese es apical oxygen
a oms and has an oc ahed al en i onmen o i s Ni2+ ca ions. We show he e ha i
he Ni2+ ions a e o ced in o a squa e plana local en i onmen as happens in he 112
ma e ials, hey p e e a low-spin d8(S=0) cup a e-like s a e ins ead, in ag eemen wi h
ecen expe imen al epo s [201].
6.4 Conclusions
In his chap e we ha e analyzed how S -doping a ec s he elec onic s uc u e and
magne ic s a e o in ini e-laye nickela es RNiO2(R= La, Nd).
We ha e ound ha including S educes he sel -doping e ec by shi ing he a e-
ea h d bands up in ene gy away om he Fe mi le el. This leads o a mo e single-band-
like pic u e, wi h he Ni dx2−y2band domina ing. We ha e also analyzed he magne ic
spin con igu a ion as a unc ion o doping (11% o 25%) and a e-ea h ca ion in he
amewo k o LDA+Ucalcula ions wi h wo di e en double-coun ing me hods. In any
113
Adol o O e o Fumega
case, a low-spin Ni2+ s a e is ound o be he lowe ene gy solu ion. Mo eo e , S -doping
highly educes he cha ge ans e ene gy be ween he Ni and O a oms.
Ou s udy shows subs an ial changes in he elec onic s uc u e o hese nickela es
upon S -doping. A cup a e-like elec onic s uc u e and magne ic s a e is ob ained when
doping hese compounds. The e o e, his esul sugges s a cup a e-like desc ip ion o he
supe conduc ing s a e ha a ises in doped in ini e-laye nickela es.
114
7 Spec oscopy o Cs2CuCl4
In his chap e we s udy he elec onic s uc u e o Cs2CuCl4. This ma e ial has
been discussed in he amewo k o us a ed an i e omagne quan um spin liquids.
A combined se o expe imen al echniques and Densi y Func ional Theo y has
allowed us o de e mine he spec um o his compound. This is mainly de e mined
by he c ys al ield spli ing ha Cu2+ ions eel when hey a e placed in he s ongly
dis o ed e ahed al en i onmen o his compound. This unpublished wo k is pa
o a collabo a ion led by D . San iago Blanco-Canosa.
7.1 In oduc ion
In he p e ious chap e we ha e analyzed he elec onic s uc u e and magne ic p op-
e ies o in ini e-laye nickela es. We ound ha when doped wi h S hei elec onic
s uc u e can be unde s ood o esemble ha o he high- empe a u e supe conduc ing
cup a es. In his chap e we con inue analyzing ma e ials wi h p omising exo ic beha iou .
Howe e , we abandon a his poin he s udy o oxides jus be o e ocusing on an de
Waals laye ed ma e ials in he nex chap e . The e o e, his wo k will se e as an in e -
lude be ween oxides and an de Waals ansi ion me al compounds. I will also highligh
he abili y o combine expe imen and heo y o de e mine he elec onic s uc u e o a
ma e ial.
Quan um spin liquids (QSL) we e hypo hesized o occu in na u e by Philip Ande son
in 1973 [202]. This is a s a e o ma e in which he elec ons’ spins emain luid-like, i.e.,
luc ua ing wi hou en e ing in a long- ange o de phase, e en a 0 K [203]. In e es ingly,
he exci a ions o a QSL, known as spinons, a e ac ional, leading o a cha ac e is ic
anspo beha iou . Geome ically us a ed spin ex u es, like iangula , Kagomè and
honeycomb la ices, a e conside ed o be he mos p omising sys ems o ind a QSL [204].
The magne ic o de and i s exci a ions will be de e mined by he compe i ion be ween
he di e en sign o he exchange in e ac ions on he us a ed spin la ice.
Among he solids showing us a ed magne ism, signi ican a en ion has been d own
o Cs2CuCl4due o he expe imen al obse a ions ha epo a wo-dimensional QSL
phase by means o speci ic hea measu emen s [205], elec on spin esonance (ESR) [206]
115
Adol o O e o Fumega
Figu e 7.1: (a) O ho hombic uni cell o Cs2CuCl4. G een, ed and blue balls deno e Cs,
Cu and Cl a oms, espec i ely. The e ahed al coo dina ion o Cu is highligh ed. The
magne ic exchange in e ac ions commen ed in he ex ha e been depic ed. (b) C ys al
ield spli ing o he Cu2+ ions in he dis o ed e ahed al en i onmen , whe e he eg
o bi als lowe hei ene gy wi h espec o he 2g. The Jahn-Telle dis o ion u he
spli s he ene gy le els wi h b1g(dx2−y2), eg(dxz,dyz), a1g(dz2) and b2g(dxy) symme ies.
and neu on sca e ing [207], which unco e ed an ex ensi e wo-spinon con inuum. The
ESR and neu on sca e ing measu emen s e ealed a weake in e chain exchange cou-
pling, J’/kB= 1.4 K, be ween Cu a oms han he in achain in e ac ion, J/kB=4.7 K,
along baxis [208] (see Fig. 7.1a). Mo eo e , in e laye coupling in Cs2CuCl4is smalle
han Jand J’ by mo e han one o de o magni ude, J”= 0.13 K, in good ag eemen wi h
he heo e ical desc ip ion o quasi-1D weakly coupled S=1/2 Heisenbe g chains [209].1
1Be awa e ha in he li e a u e an i e omagne ic exchange cons an s a e commonly de ined as pos-
i i e when dealing wi h QSL models.
116
7 Spec oscopy o Cs2CuCl4
Besides, cen al o he s udy o spin exchange in e ac ions in us a ed magne s is he
p ecise knowledge o he c ys al- ield g ound s a e symme y and hopping ene gies o he
S= 1/2 Cu2+ ions. In ac , he expe imen al de e mina ion o he c ys al ield and Jahn-
Telle spli ings, ∆CF and ∆JT has emained elusi e. Mo eo e , he o bi al occupa ion
is commonly in e ed om he oxida ion s a e in an ionic pic u e, bu is usually limi ed
in co alen sys ems, whe e he s ong o e lap be ween dand pbands gi es ise o cha ge
ans e e ec s [210]. Indeed, he mos no iceable e ec o he di e en o bi al occupa ion
is highligh ed in he manganese pe o ski es, RMnO3(R= a e ea h) [211] and cobal a es
[146], whe e he g ound s a e p ope ies a e s ongly in luenced by he si e symme y o
he ansi ion me al ion and he elec onic ea angemen wi hin he d-shell [212].
Expe imen ally, he c ys al ield spli ing, especially in 4 sys ems, has been measu ed
by inelas ic neu on sca e ing [213, 214], bu he in o ma ion is some imes hampe ed by
phonons o small amoun o c ys als. In ansi ion me al compounds, X- ay abso p ion
spec oscopy (XAS) is adi ionally used o p obe he magni ude o he c ys alline elec-
ic ield spli ings in he 3dshell [215] and he deg ee o he 3dhyb idiza ion in he
g ound s a e. Ne e heless, he mul iple e ec s a e o en hinde ed by he me al-ligand
hyb idiza ion, which smea s ou he spli ings o he 3ds a es. The e o e, a comp ehen-
si e de e mina ion o he di e en ene gy scales is usually a edious ask and elies on
se e al complemen a y echniques.
On he o he hand, band s uc u e calcula ions ha e epo ed he elec onic p ope ies
o Cs2CuCl4and p edic ed a s ong dependence on he Jahn-Telle dis o ion, ∆JT , as
a unc ion o he co ela ion exchange unc ional [216]. Howe e , a di ec compa ison
be ween heo y and expe imen is s ill missing.
In his wo k, we ha e ci cum en ed hese d awbacks by combining esonan inelas-
ic X- ay sca e ing (RIXS), Densi y Func ional Theo y (DFT) and clus e calcula ions,
which allows us o p o ide a sa is ac o y and comp ehensi e desc ip ion o he Cs2CuCl4
elec onic spec um.
7.2 Expe imen al and Compu a ional Me hods
Single c ys als o Cs2CuCl4we e g own by ou expe imen al colleagues D . F. Ro-
d íguez and D . S. Blanco-Canosa. The quali y o he single c ys als was checked by
X- ay di ac ion, esul ing in la ices pa ame e s a= 9.77 Å, b= 7.61 Å, c= 12.41 Å,
and inelas ic Raman sca e ing [217]. DFT calcula ions [17, 18] we e pe o med using
he all-elec on, ull-po en ial wien2k code [30] based on he augmen ed plane wa e
plus local o bi al (APW+lo) basis se . The gene alized g adien app oxima ion (GGA)
in he Pe dew-Bu ke-E nze ho [19] scheme was used o he exchange co ela ion unc-
ional, wi h a ully con e ged k-mesh o Rm Kmax=7.0 and mu in- in adii o 2.5, 2.22,
1.91 a.u. o Cs, Cu and Cl, espec i ely. RIXS expe imen s we e pe o med by D . S
Blanco-Canosa a he U41-PEAXIS beamline a BESSY II a 20 K and combined ene gy
esolu ion, ∆E≈150 meV.
117
Adol o O e o Fumega
7.3 Resul s and Discussion
The c ys al s uc u e o Cs2CuCl4is o ho hombic wi h space g oup Pnma. Each
Cu2+ a om is su ounded by 4 Cl−ions in a dis o ed e ahed al coo dina ion (Fig.
7.1a) which spli s he d-le els in o low ene gy 2- old egand high ene gy 3- old degene a e
2go bi als (Fig. 7.1b). The isola ed CuCl2−
4 e ahed a uni s a e u he Jahn-Telle
dis o ed owa ds a lowe symme y D2dpoin g oup [218]. This en i onmen p o ides a
spa ially aniso opic spin-1/2 iangula an i e omagne o Cu-Cl chains on a geome ical
bc plane, bonded by Cl−ions along he adi ec ion. Ou magne ic suscep ibili y measu e-
men s show no ace o magne ic o de down o 5 K, which is in good ag eemen wi h a
weakly us a ed magne [219].
Figu e 7.2 summa izes he DFT esul s o Cs2CuCl4 o he simple uni cell depic ed
in Fig. 7.1a. Only he e omagne ic s a e was conside ed, since he e is only one in-
equi alen Cu a om. Calcula ions assuming a non-magne ic g ound s a e di ec ly gi e a
non physical me allic solu ion. We ha e ound ha calcula ions on a 1×2×1supe cell
p o ide an an i e omagne ic o de as he lowe ene gy solu ion o he g ound s a e, in
ag eemen wi h he epo o Foye so a e al. [216]. Ne e heless, we ha e op ed o he
e omagne ic g ound s a e in he simple cell, since ou aim is no he de e mina ion o
he magne ic o de and does no ha e a subs an ial e ec on he elec onic ene gy le els.
The e o e, he e omagne ic solu ion allows o an easie isualiza ion o he elec onic
s uc u e and he di ec compa ison be ween DFT and he RIXS spec a. Be ween -4
and 0 eV (Fig. 7.2a) he DOS shows a clea con ibu ion o he Cu 3dand Cl 3pbands,
while he Cs bands do no con ibu e o he DOS a he Fe mi le el and, he e o e, do
no hyb idize wi h Cu. The Cu and Cl bands a e sepa a ed om he nex unoccupied
s a es by a gap o 4.5 eV, ha ing mos ly Cs cha ac e . We call his cha ge ans e gap
and co esponds o he ene gy di e ence be ween he [CuCl4]2−clus e and he Cs+ions.
No e ha i mus be di e en ia ed om he adi ional cha ge ans e gap de ined be-
ween he ansi ion me al dand he ligand ps a es in oxides [220]. On he o he hand,
he d-d gap appea s below 1 eV and ep esen s he ene gy gap be ween he Fe mi le el
and he emp y dxy band.
Figu e 7.2b del es in o he band s uc u e o Cs2CuCl4. Bonding and an ibonding
bands show up be ween -4 and -3 eV and be ween -1 and 1 eV, espec i ely. The un an-
gled elec onic band s uc u e gi es a clue abou he b eaking o he o bi al degene acy
ske ched in Fig. 7.1b. In a e ahed al coo dina ion, he 3d9elec onic con igu a ion
spli s he c ys al ield gene a ed by Cl−ions su ounding a Cu2+ ion in o he ene ge i-
cally lowe Cu eg(dx2−y2and dz2) double and highe Cu 2g iple (dxy,dxz, and dyz).
Owing o he Jahn-Telle -like uniaxial dis o ion o he e ahed on, he 2g iple is u -
he spli in o he doubly degene a ed dxz,dyz s a es (egi educible ep esen a ion) and
he hal - illed dxy s a es (b2g), and he dx2−y2(b1g) and dz2(a1g) become also ene ge ically
nonequi alen (Fig. 7.1b).
Ha ing ca ied ou a heo e ical desc ip ion o he elec onic s uc u e o Cs2CuCl4,
we p oceed wi h he expe imen al analysis o he RIXS spec a.
118
7 Spec oscopy o Cs2CuCl4
Figu e 7.2: (a) P ojec ed DFT densi y o s a es (DOS) o he e omagne ic s uc u e o
Cs2CuCl4, Cs (Cu, Cl) a om in g een ( ed, blue). Bo h cha ge ans e and d-d ene gy
gaps ha e been highligh ed. (b) Ene gy bands o he majo i y (mino i y) spin channel
on he le ( igh ) panel. The d-o bi al cha ac e o he Cu a om is depic ed as ed ci cles.
119
Adol o O e o Fumega
The as imp o emen o he RIXS ins umen a ion in ene gy esolu ion o so X- ay
RIXS has allowed o s udy he low ene gy elec onic p ope ies o co ela ed oxides, gi ing
de ailed in o ma ion o collec i e magne ic, cha ge and o bi al exci a ions in oxides [221].
Figu e 7.3b illus a es he case o he Cu2+ ion wi h a 3d9elec onic con igu a ion in a
e ahed al c ys al ield (Td), wi h a hole in an |xyis a e. In he ini ial s ep o he RIXS
p ocess, a pho on esonan a he Cu L3edge (2p→3d ansi ion) is exci ed om he
g ound s a e, |ii, in o he 3dshell (in e media e s a e, |ni) illing he |xyio bi al and
c ea ing an exci ed co e-hole s a e. In he inal s ep (| i), he co e hole is annihila ed
ia decay owa ds he g ound s a e (elas ic sca e ing, Eloss=0 eV) o an exci ed s a e
(magnons, phonons, d-d ansi ions) [222].
(a)
(c)
(b)
(d) (e)
Figu e 7.3: (a) Expe imen al (black) and calcula ed ( ed) X- ay abso p ion (XAS) o
Cs2CuCl4. (b) Schema ics o he RIXS p ocess showing he ini ial, in e media e and inal
s eps. (c) RIXS map plo ing he ene gy loss, Eloss, agains he incoming ene gy, Ein,
o Cs2CuCl4a 20 K. The maximum in ensi y o he inelas ic ea u es appea s a Ein=
931 eV. (d) Close up iew o he RIXS scan a Ein=931 eV, highligh ing he elas ic, d-d
exci a ions and he cha ge gap in he inse . (e) In-plane momen um dependence o he
d-d exci a ions, showing no o bi al dispe sion.
Figu e 7.3c displays he inciden ene gy (Ein) s ene gy loss (Eloss) RIXS map. The
maximum in ensi y o he esonan ea u es is obse ed a Ein= 931 eV (Fig. 7.3d), 0.5 eV
below he maximum o he L3abso p ion edge (Figu e 7.3a), which consis s on a ea u e-
less abso p ion band. In nice ag eemen wi h he DFT calcula ions, he cha ge ans e
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7 Spec oscopy o Cs2CuCl4
exci a ions esul ing om he [CuCl4]2−clus e o he Cs 6ss a es a e obse ed as a b oad
band a Eloss= 4 eV, ma ching he op ical gap obse ed by abso p ion spec oscopy [223].
As shown in he inse o Fig. 7.3d, his cha ge ans e gap displays 2 b oad bands a 3.7
and 4.2 eV, co esponding o ansi ions om he Fe mi le el and he dxy o bi al abo e he
Fe mi le el (Fig. 7.2b) o he uppe Cs 6sbands. The egion be ween Eloss=0.5-1.2 eV
(ma ching he ene gy di e ence be ween he 2 b oad cha ge ans e bands) co esponds
o he so called op ically o bidden c ys al ield d-d exci a ions, as widely epo ed in
supe conduc ing cup a es [221], and iden i ied in he DFT calcula ions below ≈1 eV. The
o bi al assignmen o hese exci a ions was e i ied by compa ing hei ene gy wi h he
DFT calcula ions. Conside ing ha he 3dand 3pbands o Cu and Cl a oms a e dis-
en angled in ene gy om he es o he bands (Fig. 7.2b), an expansion o he Bloch
mani old can be pe o med in eal space in e ms o localized Wannie unc ions. The
68 Bloch bands in he ene gy window be ween -5 and 1 eV ha e been used o gene a e
he localized and a om cen e ed Wannie unc ions spanning such Bloch mani old [224].
A symme y analysis o hose Wannie unc ions allows o iden i y how he c ys al ield
spli ing a ec s he Cu 3do bi als. As shown in Fig. 7.1b, we ound ha he dx2−y2(b1g)
and he dxy (b2g) o bi als co espond o he lowes and highes ene gy le els, espec i ely.
Apa om ha , he s ong o ho hombic dis o ion o he e agonal en i onmen in-
e s he ene gy le els o he doubly degene a e dxz and dyz (eg) and dz2(a1g). The e o e,
d-d ansi ions o igina e om he decay o an elec on om he dx2−y2,dz2,dxz and dyz
o bi als due o he b oken degene acy o he 3ds a es.
To be e unde s and he d-d exci a ions in he RIXS spec a, we ha e adop ed
he hole language, whe e he g ound s a e ep esen s a hole in he dxy o bi al, hence,
adyz o bi al exci a ion co esponds o mo ing a hole om he dxy o he dyz o bi al.
Wi hin he ene gy esolu ion o ou expe imen al se up, we can disc imina e he 3 o bi al
in a omic ansi ions; 2 sha p exci a ions a 0.67 and 1.02 eV, espec i ely and a shoulde
a 1.21 eV (see Fig. 7.4b). Ha ing conside ed only he e ahed al poin g oup,Td, his
would esul in a 10Dq alue o 0.45 eV. Fu he , we see no o bi al dispe sion (Fig. 7.3e)
indica ing highly localized d-d exci a ions, as expec ed due o he s ongly ionic cha ac e
o Cs2CuCl4compound. Since hese o bi al d-d exci a ions a e in a-a omic and well
localized, hey can be simula ed wi hin a ull-mul iple calcula ion conside ing a single si e
o a Cu2+ ion o he D4hpoin g oup, isomo phic wi h he D2d[225], which is exempli ied
by a egula e ahed on elonga ed along one o i s C2axes. The RIXS simula ions we e
ca ied ou by D . S. Blanco-Canosa wi h he Quan y code [226, 227] including he
Coulomb in e ac ions and mul iple e ec s lea ing ee he adial in eg als Dq(c ys al
ield spli ing), Dsand Dτ(dis o ions o he e ahed a), which a e he spli ing e ms
o he Y0
2and Y0
4sphe ical ha monics, as inpu pa ame e s in he calcula ion. Figu e 7.4a
displays he calcula ed RIXS map o Cu2+ wi hin he D4hpoin g oup. The 3 o bi al
ansi ions co esponding o he dx2−y2,dz2and doubly degene a e dxz,dyz o bi als a e
clea ly iden i ied. As shown in Fig. 7.4b, he expe imen al and heo e ical RIXS spec a
is ai ly well ep oduced a e no maliza ion o he same heigh o he d-d ea u es, wi h
an addi ional ins umen al and expe imen al b oadening o 0.1 and 0.25 eV, espec i ely.
121
