INFRARED PROPERTIES OF HIGH-TEMPERATURE

SUPERCONDUCTORS WITH SINGLE AND TRIPLE COPPER

OXYGEN PLANES

B y

Tatiana Startseva

M. Sc. (Physics) Brock Cniversitx

A THESIS SUBMITTED T O

THE FACULTY OF GRADUATE STUDIES

IN PARTIAL FULFILLMENT OF

T H E REQUIREMENTS FOR T H E DEGREE OF

DOCTOR OF PHILOSOPHY

MCMASTER UNIVERSITY

.July 1998

@ Tatiana Startseva. 1998

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11< PIZOPERTIES OF HTSC \ \ ITH SISGLE A S D TRIPLE Cu@ PLATES

DOCTOR OF PHILOSOPHY (1998)

(P hysics)

TITLE: Infrared Properties of High-Temperature Superconductors with Single and Triple

C 1 i i 0 2 Layers

ACTHOR: Tatiana Startseva. M.Sc. (Brock University)

SCPERI-EOR: Professor T. Timusk

SCAIBER OF P-AGES: s i i i , 109

Abstract

This thesis includes work on two high temperature superconducting systems (HTSC').

rianicly. La2-,Sr,Cu04 and HgBa2Ca2Cu308.+a. The C u 0 2 planes contained witiiiii thcl

rrystal structure of these materials. as well as in al1 the other HTSC. hold the kq- to sii-

pcrcoritluct it-ity. From one point of view the two systems described in t his t hesis rcprcwiit

t~vo cstrcmes within a mide variety of HTSC. La2-,Sr,Cu04 is the ciassic representiitii-O

of a material with a single Cu02 plane per unit ce11 and has one of the Iowest critical

tcniperatures. O n the other hand, the crystallographic structure of HgBa2Ca2CusOS,,, is

hsed on t hrce C u 0 2 planes. Despite the fact tha t both of these systems were disco\vwrl

quite somc time ago relatively Iittle work has been done on theni.

Irifrared spectroscopy has established itself as a very powverful technique for stuc1'-iiig

thcl properties of superconductors. One out of the vast majoriti- of excitations tliat

irifrarcd spectroscopy can probe is the excitation responsible for the pairing mecliimiçiii

i r i siipcrconductivity Furthermore. the e n e r e range covered is compatible wvit 11 t l i c ~

pi-cdictcd value of the superconducting energy gap.

This n-ork is focused on, but not confined to. the infrared study of the normal propot--

tios of bot h Lan-,Sr,Cu04 and HgBa2Ca2CuJOs+a. Tlic rcflectivity of La2-,Sr,C'iiO 1

single crj-stals has been rneasured in the frequency range 30 - 9.000 cm-' (0.004

I v\') in directions parallel and perpendicular to the Cu02 planes (abplane) . n-1icw;ts

H ~ B H ~ C ~ ~ C ~ ~ ~ ~ + ~ single crystals were studied o d y in the direction of the abpianc. h t 11

tlic cloping ancl the temperatrire dependence were studied. For the first timc. tlic iiigli-

tclrriperature optical spectra were obtained. This allowed us to investigate the c-Ii;u-go

dyrianiics of the anomalous normal state up to 400 K.

We observe a gaplike depression in the effective scattering rate l/r(w. T) b(4ow a

ttlrripcrature T* in both systems. This characteristic behavior of the frequency arid t ~ r i i -

pcratiire dependent scattering rates suggests the existence of a pseudogap statc iii 110th

t h under- and overdoped regimes of the single-layer HTSC systern La*-,Sr,CiiO 1 for

trnipcratures esceeding 300 K as well as in the underdoped regime of HgBa2Ca2Cii:10,-,j.

T h signature of the pseudogap state in the direction perpendicular to the Cu02 pla~icx

is also cliscussed.

1 would like to take this opportunitÿ to thank my supervisor. Tom Timusk. with ivliorii 1

liaïc had tlie pleasure to work for the past four years. for his invaluable insights. guiclanw.

mtliiisiasrn and patience. He spent a tremendous amount of time adrising, reassiiriiig

m t I support ing me.

1 wish to thank Prof. Jules Carbotte mho encouraged my interest and spent liis t i r i i c ~

iri discussions as weli as for providing tlie opportunity to meet and collaboratc n ' i t l i

the other physicists in Canada through series of CI-AR meetings. 1 offer sincere tliariks

to Profs. Bruce Gaulin, John Preston and Car1 Stager for their useful ad\-ises and for

pro\.itling nie with rnany unique opportunities throughout my graduate studies.

-4 warm +.tliank ?ou'' goes out to al1 the members of the Department of Physic-s i i ~ i ~ l

."ltronorni, in particular, Cheryl, Jackie. Narg. Rose and \Vend!= for their support aii(1

liclp. -4s weI1. thanks to Gord Hewitson, Venice Perno and the guys from the niachiri(. i i i ~ i

c4ectronic shops for their excellent. speedy work whicii kept the lab running srnootliI>*.

C;orci. J-our tastefiilly arranged fruit baskets and esquisite flourer bouquets will n m w Iw

fosgottcri. Tliaiiks should also be given to '- the basement community": Dimitri Basai-.

.-\iidy Duft. -4ndy Duncan: Ron Francis. Jim Garrett. .Jeff 1IcGuire. Steve '\loffat. Taciclk

Olcrli. .Jason Palidwar, ,Anton Puchkov, Toomas R6Gm. and. last but not lest. Roll

Hiiglics for ttieir huge contribution to this thesis and for providing such a dciiglirful

work atniosphere. It is mithout a sarcasm (for once) that 1 Say. it's heen great liai-irig

jwi i aroiind. My dear friends, Andÿ "the pie man" Düft and " short & snappy" Siisicl.

clcwrvc extra thanks. Discussions with them (mostly starting before $:O0 -\LI) \ w r - c >

iiltvays cnliglitcning and often led to a new avenues of thoughts.

. \ I j* gcriuine thanks and appreciation are extended to niy frie~ids: Peter and ,Ariit;i

.\Iasori. Jason and Melonie Palidwar: Ron and Sonia Francis. Frank Hayes anci Carol

R.ogcrson. The 1 s t four years wvere both pleasant and memorable.

1 an1 fortunate t o feel the love, support and encouragement of mu mother ancl rrij.

graridparents. Thank you for trying to understand what it is that 1 do and makirig r r i ~

t)clicve t h 1 could succeed in whatever 1 chose. You have bceii a wonderfully positivca

irifiiiencc.

Firially. to my perfect companion. Jim. wvho convinced me to undertake thc sr ilcl>-

at lIc.\Iastcr and filled al1 these years with the encouragement. love. faith and c:orist~~~it

support. 1 express mÿ deepest admiration and appreciation.

Abstract

Acknowledgment

Table of Contents

iii

v

1 INTRODUCTION 1

1.1 On a historical note . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 1

1.7 HTSC and conventional superconductors . . . . . . . . . . . . . . . . - . -1

1.2.1 Coupling mechanism . . . . . . . . . . . . - . . . . . . . . . . . . 4 -

1.2.2 Gap syrnrnetry . . . . . . . . . . . . . . . . . . - - . . . . - - - --J

1.2.3 .-lnomalous normal state properties . . . . . . . . . . . . . . . . . S

1.2.4 Thesis overview . . . . . . . . . . . . . . . . . . . . . . . . . . . - 12

2 THEORY AND METHODS: IR Optical Properties of Solids 14

'2.1 Rcflectance measurements . . . . . . . . . . . . . . . . . . . . . . - . - . 14

2.2 Tlicory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19

2.2.1 Dielectric response function . . . . . . . . . - . . . . . . . . . . . 20

2.2.2 Kramers-Kronig transformations . . . . . . . . . . . . . . . . . . . 2 1

2.2.3 The free electron's dielectric function . . . . . . . . . . . . . . . . -1 -1 --

2.2.4 Drude theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . - 23

2.2.5 Estended Drude forrnalism . . . . . . . . . . - . . . . . . . . . . . 2 -3

2.3 Optical properties of superconductors . . . . . . . . . . . . . . - . . - . . 30

3 THE PSEUDOGAP STATE OF UNDERDOPED La2-,Sr,Cu04

vii

4 THE PSEUDOGAP STATE OF OVERDOPED La2-,Sr,Cu0.1

5 OPTICAL PROPERTIES OF HgBa2Ca2Cu3Osia

6 CONCLUSIONS

Appendices

-4 Energy units:

B Material abbreviations:

Bibliography

List of Tables

1.1 lleasured values of A. . . . . . . . . . . . . . . . . . . . . . . . . . . . .

2.1 Combination of the lamps, beam splitters and detectors used in the LIichel-

son interferorneter. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1-1

List of Figures

1.1 Evolution of the critical temperature since the discovery of superconduc-

tiviti- by K. Onnes in 1911. . . . . . . . . . . . . . . . . . . . . . . . . . 3

1 -4 sctiematic diagram of the Cu02 plane and HTSC's crystal structure. . (j

- 1.3 Plrase diagram of La2-,Sr,CuO+ . . . . . . . . . . . . . . . . . . . . . . I

2.1 Tlic cnergy scales involved in superconductivity. . . . . . . . . . . . . . . 1.)

2 Scliernatic diagram of the apparatus used to measure IR reflcctancc. . . . 1G

2.3 Diagram of the sample holder and the cold finger. . . . . . . . . . . . . 1s

2.4 The frequency dependence of the real and imaginary parts of optical con-

ductivity given by the Drude theory. . . . - . . . . . . . . . . . . . . . . 25

2 . The real and imaginary parts of the optical conductivity of a superconductor. :30

2.G Signature of the pseudogap in the temperature dependence of the in-plaric

rcsistivity, the Hall effect coefficient and the susceptibility for t lie t hreo

rcgions of hole concentration. . . . . . . . . . . . . . . . . . . . . . . . . :Q

2.7 Tiic c-asis conductivity of an rinderdoped Y123 crystal. . . . . . . . . . . X3

2.8 Scattering rate and effective mass for underdoped cuprates. . . . . . . . . 135

2.9 Scattering rate and effective mass for optimally doped cuprates. . . . . . :37

2.10 Scattcring rate and effective mass for overdopcd cuprates. . . . . . . . . . :3S

3.1 The temperature dependence of the in-plane resist ivi ty of Lar .86S~0. L 4 Ci10.~

. . . . . . . . is shown with a sharp superconducting transition a t 36 I

The eoniparison of the seflectance of LaI.i8Sro.22Cu04 \vit h the reflectancc

calculated from the data taken on a LSCO thin film by Quijada et al. . . G4

The temperature dependence of the optical conductivity (01 ) of La1.816Sr0.1s C'IL( 1.1

. . . . . . . . . . . . . . . . . . . . . . . . (a) and La1.78Sr0.22C~0.1 (b). CjG

Tlic frequency dependent effective scattering rate of Lal.si6Sr~.18.iCu04 (a)

. . . . . . . and Lal.7RSr0.22C~04 (b) as calculated iising Equation (3.37). G S

Tlic phase diagrani T* vs x for La2-,Sr,CuO+ . . . . . . . . . . . . . . . 1 2

T h frequency dependent effective mass of Lal.81GSro. rs4Cu04 (a) and Lal .;#S I -~ , .~ : !C' I IC~

(1)) as calculated using Equation 3. . . . . . . . . . . . . . . . . . . . . . 74

The frequency dependence of the in-plane rcflectance of HgBa2Ca.rCu30s,n

is siiown for temperatures below and above Tc. . . . . . . . . . . . . . . . II)

The real part of the optical conductivity vs frequency for HgBa2Ca2Cu308,,i

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . witli ELc. $0

The frequency dependent effective scattering rate for HgBa2Ca2Cu30s,,j

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . with ELc. S2

Tlic frequency dcpendent effective scattering rate wit h E-Lc for Hg1223

(top panel), Bi2212 (middle panel) (-After Puchkov[Pucliko\-96cIj).

1-123 (bottom panel) (After D.S. Basov[Basov96]) and LSCO. . . . . . . 8-4

The area under the curve of ol(w)) as a function of frcquency. . . . . . . 86

Tlic rcal part of the dielectric function vs fsequcncy for HgBalCa2Cii308,,r

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . with ELc. SS

sii

In nature S injnite book of .srrc.rcqj

A little can I I - P ~ .

Chapter 1

INTRODUCTION

1.1 On a historical note

I r i rriany conductors the electrons responsible for the current undergo a transitiori irito

a i l ordercd state. i.e., a superconducting statc, with many unique propcrties. Tlio dif-

ftvricc between the superconducting and normal states is very small in enerp . 1)iit i t

is trcnicndous as far as the resistivity and magnetism are concerned. Both zero rosis-

tarice and perfect diamagnetism are the hallmarks of superconductivity. The nmisliiii::

of rcsistarice to the flow of electric current below a critical temperature. Tc. was c1isc.o~-

rrrd by H. Kamerlingh Onnes in 191 1 while studying the temperature dependence o f t l i ~

c.lc.ctrical resistance of mercury. The critical temperature of the first supcrconductot- n-as

-1.2 K. Tlic immeasurable resistance at temperatures below Tc suggested the naiiiv for

tlic phcrionicnon. The other key property of a superconductor is the appearaac-i1 of a

lm-go ciiamngnetism in the superconducting state. This was clisco\.cred 1- 14'. '\ Icissri(~r

m r l R. Ochsenfeld in 1933. -A bulk superconductor a t temperatures belon. T, belia\w iis

if the rriagnetic induction B is equal to zero inside the sample.

-411 csplanation for this challenging problem was not found for dccadcs. This Wils

partially duc to the fact that i t required a lot of imagination to deal with siicli drnriiatic

cliarigcs on a vcry small e n c r u scale. For esample. in the case of aluminuni thc ciic3rgy

diffrroiicc bctween an electron in the normal and superconducting state is 0.14 x lu-" PI'

I>car clcctron. This is an insignificant change coiiipared to the electron's kinetic ciici.~? of

C'h ap ter 1 . IiVTROD UCTION

1-10 c i ' . Oniy in 1957did J. Bardeen, L . S . Cooper. and .J.R. Schrieffer establish th first

siicc.cssfiiI macroscopic t heory for superconductivity called the BCS t heory. This t liw )r>-

provcs t liat it is energeticallÿ favorable for electrons t o form pairs of carriers wi t h opposi t~

spiri and momentum, via phonon mediated processes. called Cooper pairs.[Bardeeri57] Iii

ortlrr to break the pair a n energ)- of magnitude 2 4 is needed. The energ'- gap. 1. a

fiiricla~nc~ital property of superconductors, is related to the critical temperaturc I)y t l i ~

fii~idiirrirrit,aI BCS relation.

ivlicrc kI, is a Boltzman constant.

Table 1.1: bleasured values of A.

Tlie value of the gap taken from tunneling csperiments.[;\shcroftTG]

The esperimcntal gaps follow this equation very closcly nrith an accuraq- of 10%.

TiiI>I

1 H O 1 9 4 0 I H O O Y 0 0 0 v a i r

Figiirc 1.1: Evolirtion of the critical temperature since the discovery of supercondiicti~-ity 11'. Ii. Onnes in 1911. Ttic discovep of the cuprates started a new esciting era in the 1listor'- of supercoii- diict ii-i ty. (La.Sr)CuO and HgBaCaCuO, cuprate superconductors wit h one of t lio 1 o n . c ~ and highest critical temperatures are discussed in this thesis.

tlic cm of Iiigh temperature superconductors (HTSC). La2-,Ba,Cu04 lias a Tc eqiial t o

30 I\.[BcclnorzSô] In itself a jump in Tc of T K \vas not a revolutionary discover>. at t l io

t ir i ir . Howcl-cr, Cu osides (cuprates) opened up a whole new famil?. of supercoritlii(-tirig

~riiitcrials ivitli liigher transition temperatures. Since 19% otlier copper-oside suptbrc-oii-

rlit(.tors with critical temperatures above liquid nitrogen's hoiling point wcrr disco\rwil.

TIio HTSC material \vit h the highest Tc is HgBa2Ca2Cu308+s(Hgl'23) whicli wis (lis-

(.ovcrcd IF Schilling et al.[Schilling93] in April 1993(Fig 1.1). At the optimal os?-gw

c.oriccritration. Tc is equal to 133 K, a record at ambient pressure tha t remains ~ i i l i t l

t a t.Iiis dq-. Superconductivity a t such high temperatures generated enthusiasni i i i r l i c i

toc:lir~ologicnl world and attracted the attention of the public. Visions of mari?- pi-wti-

(.:il applications were proposed. Powerful superconducting magnets: magnetic rcsoriiiïic.cs

Ch apt er 1. INTROD UCTION 4

irriagirig. brain and heart measurernents with superconducting SQCIDs. long distiiiic~

Iossless transmission lines. levitated trains. lossless computer elements are just soiiirb o f

t licrri. The most promising application of the HTSC materials are passive rnicrou-;i\-c1

dm-iccs for use in satellite communications.

1 -2 HTSC and conventional superconductors

1.2.1 Coupling mechanism

T t i ~ r c arc many differences between high-temperature and coiivent ional superconrliic-t ors.

Thc major difference is the magnitude of the critical temperature. Often. the T, o f t h

ciipratcs is almost an order of magnitude larger than the Tc of conventional supclrroii-

diictors.

Tlic s t rongly-coupled BCS t heory describes convent ional superconductors ver!. l \ - ~ 1 1 .

Plionon-rncdiated pairing is responsible for the superconductivity in tliese matcrials. Tliis

\vas pro\-cn by the isotope effect. In conventional supcrconductors Tc is proport ioiial t o

thc frqiiency of phonons which is in turn inversely proportional to the m a s .

For s~ipc~rconductors such as Hg, TI and Pb a = 0.5.[I

C'liap ter 1. INTRODUCTION - il

typcs of copper-oside layers commonly referred to as layers and chains. and up to a

C'2lapter 1. INTRODUCTION

CuO, plane

G

Figure 1.2: -4 schematic diagram of the Cu02 plane and HTSC's crystal structiiro. Thc upper figure shows the planar arrangement of Cu and O atoms. Cu atoms hav(3 riiagnctic moments which form an antiferrornagnetic sublattice. The lower figiirr is iiiost1~- based on (La.Sr)CuO's crystal structure. It empliasizes the conducting CuOZ Iayclrs separated by a couple of insulating planes. The direction of CuO? structure is c-allcrl t hc a b-plane. The perpendicular direction is called the c-axis.

bctiwen d and s mave symmetry is not just in the shapeof the order parametci-. I ~ i i t

also in tlic fact that d-wave has a sign change associated with it. In k-spacc tiio orch'r

I)xrarricter is related to the energy gap, A. The particular direction in k-space ~vlici-(1

thc gap \.anishes would ultirnately be reflected in the experimental data as the prcs(wc*c.

of rscitations starting at zero energ-y for the esperimental techniques that represciit aii

aïcniged ovcr al1 (or the majority) k space. Such esperinients include infrared (IR) spot--

troscopy. microwave conductivity, XMR and specific heat measurements. For es~irriplc~.

one of the first esperiments to suggest d-wave symmetry in the order parametcr wiis

Figure 1.3: Phase diagram of La2-,Sr,Cu0.1. Thc gcneral features, such as the presence of the antiferromagnetic phase and tiio stipcrcondiicting state: are present in al1 HTSC.(from Ref. [Schilling93])

tlir ineasurements of the linear-in-temperature(as opposed to T2) penetration deptli in

\-Ba2Cii306.Bs (Y 123) by Hardy et al. [Hardy931 This shows t hat the density of cscitat ions

clrops more slowly with temperature than it would for a constant gap.

Tlicrc are several esperiments that can actually resolve the k dependencc of thcl g i p

s>-rrirnctrj-. Polarization-dependent angle-resolved photocmission studies by 2.-S. S1ic.n

a i t I Campiizano's groups on high-quality untwinned single crystals of Y123 and Bi2C;iCii2( )+,,

(Bi-212) rcveal modifications to the near-Fermi-edge spectral weight that are indicati\-r

of ari anisotropic superconducting energy gap.[Shen93? Harris96? Ding96. Ding971 Tlir

~rirasiircd magnitude of the shift of the leading edge increases monotonically, reachi~ig ii

rriasimum value of approsimately 25 mcV. The authors note that within esperinic31it;il

Chap ter 1. IXTROD UCTION S

rrror tlicir results are consistent with a dZ2+ order parameter. where ~ ( k ) is propor- 4

tional to cos(k,,) - cos(&): but qualitatively incompatible with an estended s-wavr gap

proportional to cos(zz,) cos(^,,,). Hoivever. they point ou t that the' cannot distirigiiisli

Ixtween pure dZ2+ symmetry and mixed s + id symmetry because they arc urial)l

C'tiapter 1. INTRODUCTION 9

Since tlie discovery of HSSC, i t has been noted[GurvitchSï] that normal s ta te resistivit~-

is liricar up to estremely high temperatures. -4 tremendous amount of experimental work

c-oriccntrating on the normal properties of HTSC has been collected o\+er the Iast t liroo

!-o:irs. Howver, there is still no consensus about the mechanism behind these uriiisii;il

phionienon.

-411 S l I R study of Y123 was the first esperiment showing the so-called "spin gai)" or

~pcwtlogap" . [lVarren89] -4s a function of temperature, both the spin suscept ibilit>- arid

t lic riuclear spin-lat tice relaxation rate was siippressed below a cliaracterist ic tcmpcra t iirr3

Te. a tcmperature that can significantly esceed the superconducting transition tcriipcBr-

atiirc T,. The spin susceptibility is proportional to the number of electrons with spiris

aligncd dong the applied field. It takes more energy t o flip tlie paired spin ratlicr tiiaii

a single spin. Thus, the spin susceptibility is suppressed. This argument producc(1 r 110

illeil t l ~ t tlie preformed pairs already esist in the normal state. The name .pseiitlogap'

clci-eloped because. first of all, it was observed above Tc and. secondly. an actual ztxro

\-;iIiio i r i the gap was never observed.

Tlic normal state transport properties of HTSC arc strongly affected by t he prcw~ic*c~

o f tlic psciidogap.[Batlogg94' Bucher93, Ito93, C\alkes93] The resistivity of l'Ba2Cii:i( 1: is

a li~icar fiinction in temperature wcll abovc 300 K. Howver . if tlic osygcn conccntrat ion

is rcdiiçcd. ttic slope of the resistivity becomes steeper dcveloping almost a T2-k>eIi:i\-ior.

Tlir tcmpcraturc a t which the slope of the resistivity changes is called Tb. Ttic \.;iIiic~

o f T* as n-cl1 as its doping dcpendencc suggests a common origin bctween the traiisporr

~)liorior~icriori and the XMR data. The spin fluctuations responsiblc for a supprc~ssicm

o f ttic Kriigtit shift could be the source of the reduction in the scattering seen i i i t l i ~

rosisr ivi ty data. On t hc o t her Ilando the specific k a t measuremcnts[Loram93. LoriiiiiO-l]

slion- that tlic rncchanism behind the anomalous normal properties of HTSC is not lirriitd

to i l xiiagnctic origin but also has a charge origin.

Several groups have studied the k-space dependence of the pseudogap using ariglc-

r (wlwd photoemission spectroscopy (ARPES) .[h;larshall96a. Loeser96. Ding96. Dirig97]

Ttir pliotoemission spectrum gives information about the electronic energ- as a furirt ion

o f k. For a normal metal a t zero temperature the spectrum is constant down to thr Ftmii

crierg_\-. CF. Below this energy a sharp drop develops. If there were a gap at the F(miii

siirface tlien the density of states would descend for energies Iess than EF. In undercto~)cd

Bi2212 it n.aç found that the drop below the Fermi energy occurs abovc Tc. The siw o f

tlic pscudogap was independent on the doping level. It also seems to dissappear a l ) o v ~

opt irrial doping.

Several ot her techniques were used to study the pseudogap. \;acuurn t uriric~liii~:

s~>ectroscopu[Renner98]: Ranian spectroscopy~[Blumberg9ï~ '\'emetschek97] and nrii t rori

scattering esperiments[Rossat-hlignod91: TranquadagP] have al1 shown evidence of a g a p -

likc featiire in the normal state of various materials.

Thc infrared reflectance technique is also suitable for investigating both tlic Iior-

i r i d and supercoriducting properties. The term "pseudogap" \vas first introducc(1 1))-

C.C. Hoi~ics wliile studying the c-a is conductivity of \ l '1S. [HomesSSb] A. \*. Piiîliko~.

rt al. [Puclikov96d] reported studies on the electromagnetic response of tlirce diff(wiit

fiirriilics of HTSC (Y123 and Y124, Bi2212. and T12201) wliicli togetiier allowcd tlio iiii-

t fiors to cover the entire doping range from under- to overdoped. T h evolution o f t l i c x

~)sc.iidogap response by changing the doping level. by varying the temperature from :i\m-r

to 1)clon. Te. or by introducing impurities in the underdoped compound {vas csplorcd.

-A iiicrnory-function analysis of the ab-plane optical data most clearly rewaled thc i+fwt

of tlic pscudogap. I t was found that it occurs in underdoped samples. One escq)rioii

to ttiis riilc \vas the compound Bi2212 where T' \vas highcr than Tc in the optiiiia11:-

rlopcd rcgime. The main conclusion was that the scattering rate of underdoped HTSC is

liricar and temperature independent over a significant frequency range (750 cm-' 111) t o

3000 cm-'). Below 750 cm-' the scattering rate drops off faster. almost as u2. The rrgiori

~ I i e r c strong suppression of the effective scattering rate occurs is called the "pseuclogap'-.

The vast niajority of the esperiments described above were done mostly on two ~ n a t ~ -

rials: Bi2212 and Y123. The technical aspects of working with Y123 are better devclopcd.

Tlic latter's critical temperature of 93 K allows an esperimentalist to perform the st i lc l>-

iisi~ig rclat ively inespensive refrigeration equipment. In addition. the niet hods of gro-

irig and working with Y123 are well advanced. Xevert heless, t hc casily cleaved siirfi~w

liai-irig atoniic smoothness made Bi2212 the material of the choice for the turinclirig ; m l

ARPES spectroscopies. However. there are some drawbacks to t hc measurements cIoric

ori Bi7212 and Y123. Both systems possess two Cu02 planes per unit cell. This ~ x i a k ~ s

aii csplanation of the role of the CuOz plane in HTSC estremely difficult. In aclrlit ion

1-123 lias a more compiicated structure because of its Cu0 chains. It has been clotor-

rniric from DC resistivity. [Gagnon941 penetration dept h measuremcnts. [ZhangIiS-l] arid

t lie IR data [Basov95a] that both the C u 0 chains and the CuO? planes bring an ~ < l i i : i l

ro[itrit~iition to the conductivity. This makes the study of the role that the Cu02 plmc~s

p1a~- in superconductivity more complicated. Furthermore, 1-123 arid Bi22 12 are 1i11ii t cd

as h r as dopiiig possibilities are concerned. In order to obtain a gcneral picturc a l i o i i t

tlicl ciopirig dependence of the pseudogap it is necessary to include different fanii1ic.s o f

HTSC.

LSCO is an escellent prototype system for studying the doping dcpcndence. \\-itIi

Sr sii1)stituteci for La, it is possible to go through a complete doping range frorri iiricI(~r-

t o ovcrtloped. In contrast, Y124 is invariahly underdoped. Y123 and Bi2212 ~ i 1 1

ii~itlcrtlopcd and slightly overdoped. and Tl2201 is always overdopeci. llorcovcr. tticx fiict

tliat LSCO consists of only one Cu02 plane per unit ce11 helps to pinpoint the rolc o f tlicl

plarics u-i t hout complications due to chain-plane interactions and the prcsence of c.loscl>-

coiiplcd Cu02 planes.

C'liapter 1. INTRODUCTION 12

Ariotlier aspect of LSCO should be rrientioned. Based on ?&IR results t h p w i -

tlogap has been attributed only to cuprates with more than one CuOp layer pcr iiriit

~~cll.[.\Iillis93] -4ccording to the authors, the weakly coupled Cu02 planes in LSCO ivoiild

prcverit long-range spin density wave ordering because of the low dimensionality This i~ii-

plies tliar, a spin-gap or a pseudogap cannot be observed in LSCO. However. the traiis~)ort

iricasrirernents on LSCO indicate anornalous normal properties similar to thosc foiriid i ~ i

1.123. r\dditionally. the IR optical rneasurements by Cchida et al.[Cchida96] contirriid

tliat a suppression of c-axis conductivity similar to Y123 was present in undcrrlopcrl

LSCO with s = 0.12. This, along with the several reasons mentioned above. stimiilatctl

our stiidy of IR opticai properties of LSCO.

--!part from the fact that the data presented in this thesis for the Hg1213 cornpoii~i(l

iiïr the first optical spectra for this material, the beliavior of the pseudogap is estrcwidy

iritrigiiing because it has a triple Cu02 layer structure. In al1 othcr measurements tlio sizc

o f tlic pseudogap in the direction of ab-plane was both material and doping indcpciirlvrit.

\\htiIrI it hold for the highest-yet-found HTSC described in this work'?

1.2.4 Thesis overview

T l i ~ rriairi focus of this research is the studi- of pseudogap as a function of dopirig. rcSlli-

prratiirc and the number of Cu02 planes by mcans of infrared optical spcctroscopy. Tiiv

c-lioicc of infrared spcctroscopy as a primary tool for this study is a rcsult of a comproiiiiso

l)c~\vccri a desire to investigate t hc broadest range of properties and iising a sirigl(~ PX-

p1rirricntal technique. It permits one to study such phenornena as electronic transit ions.

pliorioris. pscudogaps. superconducting gap, and the scattering rates of conductirig c-iirri-

m-s i i i t h cnergv range from 10 cm-' u p to 10000 cm-'. This frequenc- range allons o r i ( l

to cstablisli a connection between low-frequcncy esperiments. for esample DC traris1)ort.

trii(:roivavc. tiinneling, Raman, to the high-energu measurcments, such as -4RPES i i i i ~ l

Cllap ter 1. INTRODUCTION

S - r v spcctroscopy.

Cliapter 2 will provide the necessary background for the analysis of the optical data

as wcll as some details of the experïmental apparatus.

Cliaptcr 3 will discuss the optical properties of the underdoped LSCO. A comparisoii

n-ith the double layer cuprates is made. Pseudogap features along both the ab-planc. a11d

cm-asis direct ions are presented.

Cliaptrr -i deals with the widely helieved statement that the pseudogap is a sigii;itiirr

o f only the underdoped families of HTSC. The manifestation of the pseudogap in ovcDr-

dopccl LSCO is described and compared to the recent data obtained from an identiwl

saniple i~sing Raman spectroscopy.

Cliaptcr 5 provides new insights into the charge dynamics of the triple layer HTSC'

Hgl2Z3.

Chapter 2

THEORY AND METHODS: IR Optical Properties of Solids

2.1 Reflectance measurements

Irifrarcd spectroscopy has established itself as a ver7 powerful technique for studying t iic

propcrtics of superconductors. One out of the vast majority of excitations that cari t ~ c .

piol)ccl by infrared spectroscopy is the excitation responsible for the pairing mecli;uiisiri

i ~ i superconductivity. Furthermore. the energy range covered is compatible witii r i i ~

lmdictcd \.alue of the superconducting energy gap as well as many other interactioris iri

tlic HTSC systems. Figure 2.1 exhibits some of the important energ-- scales in rwtals

i i ~ i < l sirpcrconductors.

T h instrument used in the study is a Fourier transform 1Iichelson interferorricltt~r.

Tlic rcflcctancc obtained in these esperiments usually corers the range froni 20 cm-' I I I )

t o 9.000 cm-'.

Siiicc the information obtained is only the real part of the complcs rcflectanc-t riicw

Iiriirncrs-Kronig analysis must be used to extract both parts of the comples o1)t i c ï i l

Ehir 2-1: Combination of the lamps, beam splitters and detectors used in tlie ~ l i c l i (~ l so~ i i~iterfcronieter.

Ri11lg~' cm- ' 20 - 250 100 - 800 - - . . -

/ 450 - 9.000 Tungsten KBr 77 I< HgCdTe(A1CT) 1

Lamp Hg arc Hg arc

GO0 - 8,000

Beam splitter i2.+(50G) klylar

3p(12G) Mylar

Detectoi 1 1.2 K Si bolom

Chapter 2. THEORY ICIETHODS: IR Optical Properties of Solids 1 5

constant. The mathematical aspects of the analysis are discussed below-. Howevcr. i t

is important to mention here the necessity of the high-frequency data in the Krariicrs-

lironig calculations. Obtaining this data requires the use of a grating spectrometer ancl

x i cllipsonieter (Fig. 2.1). Besides supplying the information for the high-freqiiriiq-

rstrapolation. ellipsometry allows one to obtain both the real and imagina- parts o f

t l i r optical constants in the near IR, visible and near ultraviolet regimes directly froiii

the cspcriment. It is useful to check the accuracy of a Kramers-Kronig calcularion Li>-

camparing it to the ellipsometric data. The avaiiable ellipsometric apparatus. howwr . is

liriiitcd to room temperature measurements and the programs are for isotropic sariipl(s.

Figure 2.1: The energ': scales involved in superconductivity.

-4 prototype of the interferorneter was developed by hdichelson in 1880. Despitcl its

age it has stiIl been used as the primary device in infrared spectrornetry. A sketch of t l i ~

espc?rimcntal setup is shown in Figure 2.2. Such instruments allow one to meastirr t 1 i c ~

Chapter 2. THEORY - A N D METHODS: IR Optical Properties of Solicis

Concave 'L Mirror w

Concave

Concave Mirror r-

Mirror R e f e r e n c r b d?* Evmporator -

Cryostat

Light Bource

Ed lovible lirror

Figiirc 2.2: Schematic diagram of the apparatus used to measure IR reflectancc. Ksi ially t hree different detectors are iised.

Chapter 2. THEORY AND METHODS: IR Optical Properties of Solids I I

irifrared refiectance as a function of both frequency and temperature. The broatl I~tir i (1

light source used was either a mercury arc lamp or a tungsten lamp. The light frorii th(.

source is directed by a mirror ont0 a mylar beam splitter which has the property of lwirig

ahlr. to reflect and transmit equal amounts of light. The two beams di\-ided by t hc I)cvirii

spIitter are reflected back by one stationary and one movable mirror. The recorri~)ixitd

Immis will t hen interfere with each other. Whether the interference is construct ivcl or

dwtriictivc dcpends on the path difference and the frequency of the liglit. After passirig

a set of rnirrors the beams travel through a chopper that is only useci in the aligriiilcwt

proccdure. Finally, the light hits the sample positioned inside of the cryostat.

The sample is glued to the apex of a cone (see Figure 2.3). To utilizc the wliolc

sariiplc. irri overfilling technique is used so t hat light t hat misses the sample is scat t t~rcvl

oiit of the optical path.[HomesSSa] The reflectance of the sample (R,) is compnrtd to

the rcflcctance of a stainless-steel reference mirror (&). By alternating betx-ecii t l i ~

s:irriplc and reference mirror every few minutes one can eliminate the effects of long t m i i

drifts in both the detector response and in the light source. The rotation also allows o r i ~

t o place the sample in front of an evaporator- To correct for the sample size aritl iiiiy

irrcg~ilaritics in the surfacel and to eliminate the efkcts of the reference mirror. ; i r i i 7 1

. s i t u o~iporat ion of a metallic film (.Au or -41) ont0 the surface of the sample \vas iisocl.

Tlic coated sample was then remeasured (Rg , ) and the absolute value of R \.as givoii I)y

tlic ratio of spectra before and after plating, corrected fur the absolute reflectance of t i i o

Tlio iicciiracy of the absolute reflectance is estimated to be < &lx. T h l)r,ass conc holding the sample was anchored witli a copper braid to the colcl fi1igc.r

of a contiriuous flow cryostat to insure proper thermal contact. The availablc tcrnpcratiirc

Cliaptcr 2. THEORY -4!VD ibIETHODS: IR Optical Properties of Solicfs

cold finger

Figure 2.3: Diagram of the sample holder and the cold finger. Strige 1: The light from the source hits the samplc and the reflected light is detectrd. Tlir rotation of the cq-ostat allows one to compare the reflectance of a sample to t l iat of a rcfercncc rnirror. Stage II: Similar measurements are repeated with the thin laycr of gold evaporated on the surface of a sample. The ratio between the data obtaincrl frorri Stage 1 and the data from Stage II gives an absolute value of the sarnple-s rcflectance.

C'liapter 2. THEORY .AND METHODS: IR Optical Properties of Solids

rarige varies froni 10 K up to 850 K. However, in the present s tudy the highest rneamrc4

tclrripcrature was 400 K.

I t is also possible to measure the reflectance d o n g different c~stal lograpli ic dircc-

tio~is. This can be accomplished by introducing a polarizer at the chopper position. Thc

ori(1ritation of the chosen axis can be calibrated by plotting the reflectance ratio as a

fiiiictioii of the polarizer angle. and then least-squares fitting it to a sin2 B functioii. T l i ~

(\stirliatetl crror in the anglc between the chosen ~ x i s of the crystal and the polarizrr asis

is Icss t h n 1". The polarizer lealcage of other polarizations is less than 0.4% of tlic sigrid.

=\ftcr passing the cryostat the light is collected by a toroidal mirror and is foc.iiscv1

orito a detector. The plot of the detector signal as a function of mirror position. wliir41

is rcfcrred to as the interferogram. can be Fourier-transformed into the light signal as a

fiirict ion of frequency.

2.2 Theory

111 order to interpret the reflectance datao the electromagnetic theory of light stiotild hc

iiscti to derive the cornples refiection coefficients in terms of the macroscopic opt i c d

properties that characterize the specific sample iinder study. .A considerable ainoii~it of

work has bccn donc on this topic. A detailcd discussion can be found in a nriri1lx.r o f

I)ooks iirid articles.[.4slicroft7û, Kittel86. Tiniusk89. \\:ooten72. LandailS.II Tlic piirpos(~

o f this cliaptcr is to provide the background information that is essential for tlic ailal!-sis

o f tlic r~ficctance da ta where the angle of incidence is ciose to normal.

Cliapter 2. THEORY AND METHODS: IR Optical Properties of Solids

2.2.1 Dielectric response function

Coiisider a mcdium which is isotropic. linear in response. and hornogeneo~s.~ If t h img-

iictic field does not va- with frequency (static case) then t h e definition of the dielwtric

wiistant E is described by the following espression:

where e is the static dielectric constant' 6 is the displacement and Ë is the electric tidd. In the dyiiamic case' i.e. when E = C(W) Eq. 2.2 would transform into.

Tlic reflcct ivity defined in terms of the dielectric function is:

Thc? cornples index of refraction, N ( w ) . is given by:

Finally. t lie power absorption coefficient is:

'The imits used in this thesis are CGS

Cliapter 2. THEORY .4!VD METHODS: IR Optical Properties of Solids 21

The physical meaning of the imagina- part of the dielectric function is it dcscritws

lion- much energy is dissipated.

2.2.2 Kramers-Kronig transformations

Tlie Kraniers-Kronig transformations relate the real and imaginap part of the corriplt~s

rrspoiisc a t al1 frequencies. However: certain conditions have to be satisfied. Co~isirlw-

t l i ~ rrsponse function f (*} = Re f (w) + i Im f (;i).[Kittel86] First of all. the poles of f (--) ~ i i i ~ s t hc bclow the real a i s . Secondly, the integral should vanish around an infinitcl

s~tiriicircle in the upper half of the complex w-plane and f (w)-+O uniformly as l i l ix-

Firiall~.. and most importantly the response must be linear.

The Kramers-Kronig relations for f are:

Hcre. P means "principal part".

T h e Iiramers-Kronig transformations are a direct consequence of the causalit>- [>riri-

c-ipkl. Re f iind I m j rnust be related to each other to satisfy causalitj..

Applyirig Eq 2.10 to the dielectric function one gcts the following:

wlirtrc ad, is tlic DC conductivit-

Firiallj-. the comples reflectance is defincd as.

Cllill>tcï 2. THEORY AND METHODS: ZR Optical Properties of Soiids -1 -- .)

In a n esperiment only reflectance? R, is obtained. The Kramers-Kronig relations iii.il

i~sed to estract the phase shift. Therefore, it is important to gain esperimental inf~rriiii-

tiori about R over as wide a frequency range as possible. Typically. the extrapolation is

iiswl to estend the da ta beyond the measured frequency interval. .At high frequenîirs a

R-d-P relationship with O 5 p 5 - l is used. For an insulator the reflectance is consitlcr(d

to be constant at low frequency. For a metal the reflectance follows the Hagen-RuI)(ws

rclatiori. R = 1 - ( h p d C / i i ) '/'! n-here pd, is the dc resistivity.[Timusk89]

Aftcr putting together the da t a and tlie extrapolation the Iiramers-Kronig relatioiis

(-;tri 11c applied:

By knowing both r ( w ) and O(o) tlie rest of the optical constants c m be deterrriiri(d

fiorri t lie Fresnel equations.

2.2.3 The free electron's dielectric hmction

Csirig the cquation of motion of a free electron in a n electric field one can get an c s ~ > r < ~ s -

siori:

Bi. siil)stituting the following equations.

irito Eq 2.15 one can obtain the expression for the dipole moment of the electron:

C'tiapter 2. THEORY =I1VD METHODS: IR Optical Properties of Solids

Tlicri the polarization which is the dipole moment per unit volume is equal to:

The dielectric function in t his case is given b_vt

or 11'- substituting Eq. 2.19 it will take the form of

Tlw plasma frequency up is defined by the relation:

Csiially. the contributions from high-energy interband processes and atomic corcs gi\.cl

risc to a background which can be included as ~ ( D c ) . Eq 2.22 would non- look likc:

Sotr t ha t orle of the ways to determine a plasma frequency is to find the conditions n-tirw

the diclectric fiiriction is zero.

2.2.4 Drude theory

Frcqiicntly. particularly in the case of metals. the comples optical conductivity a(;) =

ol ( r i ) + io2 (LJ ) is considered. The dielectric function and the optical concliicti\-itj* arcb rvlatccl hy the equation:

C'hapter 2. THEORY AND IUETHODS: IR Optical Properties of Solids 24

P. Drudc established a simple rnodel of metallic conduction in 1890.[.-\shcroftTGj Tli. is still used as a starting appro-ximation in complicatcd situations. Tlicrr i w

sc3vcraI assumptions that have to be made when applying this rnodel. The first is tlio fr-cv

cIcctron approximation. The second assumption is that the collisions occur betweeri tn-o

or more clectrons and static defects instead of between electrons or between eh-troris

ii1ic1 other excitations such as phonons and magnons. The probability of a collisiori p r

iirtit tinic is given by Z / T . The third assumption states that the average time bcltn-rrm

mllisioris is independent of position and velocity. Finally. it is assumed that elwtroiis

a(-hicvc t hcrmal eqiiili brium only t hrough collisions.

Coriçidcr that al1 the above assumptions are satisfied and under the influence of an

csternal field the Fermi surface moves in the opposite direction to the field by a n airioirIit

proportional to current density. The equation of motion of an electron with monic3ritiirii

17 clspcriencing a periodic electric field is,

Siil>stitiitiiig Ë = Ëo exp(-iwt) and = esp(-iwt) one can obtain

Tho ciirrmt density is,

Corisccliicritiall_v, the optical conductivity in the Drude mode1 takcs the form:

Ctiapter 2. THEORY AiVD METHODS: IR Optical Properties of Solids

~vlicrc O,!, = ne2r/m is the d c conductivity.

T!ie rra1 and imaginary parts of the Drude conductivity can be presented as

Hcm. ;, = 4iine2/rn is the plasma frequency of the free carriers. A typical a n S e o f t l i c rcwl corictuctivity @-en by Eq.s 2.32 and 2.33 is shown in Fig. 2.4.

Figiirt: 2.4: The frequency dependence of the real and imaginary parts of optical coticiiic-- t i\.ity gi\-en by the Drude theory.

2.2.5 Extended Drude formalism

111 rcal systems such as HTSC, the conductivity has a more elaboratc form thari tliiit

dos(:ril~ccl by a simple Drude model. Two models have been used for HTSC: oiw i i ~ i < l

tn-O corriponcnt models. In the one component model. the optical responsc is diic t o

;il)sorption of energ'; by one source of carriers. the itinerant conduction electrons. n-it l i a

Chnptcr 3. THEORY AND METHOLE: IR Optical Properties o f Soiids 2G

frcqucncy dependent scattering rate l / r ( w ) and a frequency dependent mass n ~ ' . This

is n result of the strong interactions with dynamic excitations arising [rom the stroiig

(ktrori-elcctron interactions. The single component formalism is coinmonl~ referrcd iis

t l i ~ ~stcr ided Drude model. For the two component model. one assumes that tlirrcl ;in1

two contributions to the frequency spectrum: one comes from the conduction cIcc-t roiis

iirid the other is related to the polarizable bound-state electron (mid infrared) carrims.

Tlic tivo coniponent niodel can also be regarded as arising From a double relaxation

procbcss. two different scat tering mechanisms. rat her t han two sets of carriers. [.-\llt~riiG!

Both models have their advantages and drawbacks. For HTSC, the spectra obtiii~icd

frorn lightly doped cuprates reveal distinct and ~7ell separated absorption features n.dl

srparatcd frorn the Drude band.[Timusk89] Thus. a two component model srcnis ap-

propriate. The lom frequency Drude band dominates more as the doping iricreascs i irid

tlit. siiigle-component approach gives better treatment of the optical response at lo\v Frc-

qiicricics bclow 1 eV. The effective mass loses its pliysical meaning and becornes ncgativcl

:il)o\-c 7.000 crn-1[Puchkov96c] suggesting the influence of additional excitations.

I t lias bcen suggested that it is safe to a p p k the single coniponent model to aiialyzt tlio

dii t a at cncrgies below 4.000 cm-'. [BasorgBa] Thomas et al. [Thomas881 noted t l i a t t lio

riiirrib(>r of carriers estimatcd from the optical spectra corrcsponcis to the value o h airiml

froni t h chernical valence argument if a ciit-off frequency of 5.000 cm-' is used. Hoiic-(B.

thcl response of conducting carriers is dominant up to 4.000 cm-'. .Uso. by csariiiriiiig

t hcl tcnipcrature dependence of the reaI part of the conductivity and considering tlièit t Ii(1

t oiripciratiire dependence is usually assigned to "free" carriers. one can conclude t h r liv

oiiv coiiiponent model can be used up to 3.000 o r 4.000 cm-'. Furtliermore. tbeor(lti(.iil

C'iiapter 2. THEORY ,Al\-D METNODS: IR Optical Properties of Solids

rrDsporise cornes from intraband processes.

However. the main deficiency of the estended Drude model is the fact that it is h s ( ~ l

or1 an ovcrsimplified version of Fermi liquid t heo r . This theory is highly questional)l(t in

HTSC. One way to avoid this shortcoming is to calculate the conductivity. first. by iisiiig

a more complicated anisotropic theory with a k dependent Fermi velocity and then iisi~ig

t lie caIciilateci conduct ivity t o find the effective scat tering rates and the effective niasstls.

Tliis approach WLS used by Branch et al.[BranchST] and Stoj korié et al. (Stojkovii.9~'j

Dcspitc the possible disadvantages of the estended Drude model it has been n-idr:~-

acccptcct in the infrared comrnunity as an estrernely fruitful tool to analyzc the c-liargr

cl>-rianiics in cuprates. [Thomas88, Collins90. Rieck95. Basal-96. Puch kov96aI -An ov(tr\.i(w

of tlic model is offered below based on the work of several theoretical groups.[-Ult~i71.

DolgovS5. Sliulga91. Gotzei2. Allen76, Webb861

Iri the estended Drude model the optical conductivity can be described by niakirig

tlic cIamping term in the Drude formula comples and frequency-dependent: -11 (;. T ) =

l/i(i. T ) - iwX(w, T)' where M(w) is called a mernory function.

1 4 1 2 Ld- ~ ( L J , T ) = - - - - P

4 1 ( T ) - z 4 i i l / r (w . T) - iu[1+ X(w. T ) ]

To rc(:o\-cr the classical Drude result the rnernory function can bc espanded for- sriiall

frcq~icncics into a Taylor series. The conductivity will then take the form:

1 .2

The alternative way t o rcduce Eq. 2.34 to the familiar Drude form is to iritrocliit-th

i i rcnormalized scattering rate I / r * ( w : T ) = l / [ r ( u . T ) ( l + X(ü . T))] and the effri-ti\-(. ~>liisrrla frcqucncy wi2(w. T) = w:/(l + X(w. T)):

Cllapter 2. THEORY AND METHODS: IR Optical Properties of Solids

L / T * ( ~ ) is physically meaningless. a t l e s t mhen interpreted as a scattering i - a t~ .

Eqiiation 2.31 demonstrates that l/r(w) and l/+) are identical escept for a r r i i i l t i -

plication factor. Thus. it is the real part of a physical response fiinction. I/o(&*). i l ~ i ~ i

olwys a Iiramers-Iironig relation. l/r'(w) does not obey a Kramers-Kronig relation i i i i~ l

is ~iot directly a physical quantity. If a theoretical calciilation of electrons scattcriiig

off of some other boson-like excitation is made, then l/r(w) is closely related to ttic

irringiriary part of the self-energy and the quantity conventionally n-ritteri izm' / r r r is

the real part[Dolgov95]. Within such calculations. in the limit of zero freqiiericy r l i c

riorrnal-state optical conductivity is completely real and is equal to the dc conductil-ity.

l /odc(T) = p d c ( T ) = me/(~(T)ne2): where pd, (T ) : is the dc resistirity. l/r*(d) C ~ O P S

riot cntcr at all. ,An alternative explanation is that the quantity with a clear ph'-sic-al

riicariing is the mean free path. IVhen one tries to convert the latter into a lifetinic. i r is

iicwssary to know the velocity and therefore the m a s . The way to separate thcscx two

offrcts is to consider l/r(w). l/r*(u) is the mean free path mised iip with the v~loci t>-

rclrior-riialization. while l/r(u) is the mean free path.

The rnass cnhancement factor X(w) is given as the imaginary part of l/o(s):

Ttic vcq. narrow Drude peak is a consequence of the large rnass. which niakes it hart1 for

carriers. even scattered ones. to stay in phase n-ith the driving field. The total pliis~~iii

fr-rqii(mcy i; in E C ~ S 2-37 can be found from the sum rule Io o,(w)du = ui/5. TIic cletailcd calculations in the frame of memory function andysis were devclopc(1 for

olcctron-phonon scattering.[;\llen76~ Allen71: Shulga91] In the case of finite tcmpcrittiirw

Stiiilga et al. obtained an espression for the effective scattering rate:

Cliap ter 2. THEORY AND METHODS: IR Optical Properties of Solids

n-tierr nK(R) F ( R ) is a phonon density of states weighted by the amplitude for large-iiiiglrl

scattering on the Fermi surface and T is measured in frequency units.

Eq. 2.39 giws an interpretation of the suppression of l / r . At ION- frequencic-s t h

iiiiriil>rr of encrgetically accessible states is small. -4s soon as the scattering ~ l i i i ~ i ~ i < ~ l ~

I)rc:orne avaliable the scattering rate increases. Yet . one would cspect the satiirat ion

of tlic scattering rate as soon as additional channels are no longer present. Farnm)rtti

mid Tirniisk were among the pioneers in the inverting the spectra to obtain n 4 F iri

BCS superconduçtors.[Farnworth74] In strong electron-phonon couplingo e s c i t d oloc--

troris cniit phonons and produce a spectrum of conductivity related to a 2 F ( ; ; ) . TIic

katrircs observed in the superconducting state are similar to that of the normal statc

( w q l t for a shift of twice the gapo i.e. " p h + 2A. The results correlate well wirli t l i v tiirincling data. 'vloreover, the optical data carried more information. -4s sccn frorxi PX-

pc'ri~nents the effective mass is strongly enhanced. This is a consequencc of t h enliiilic-cd

sc:at tcririg.

I t sliould bc clear from Eq. 2.39 tliat the extraction of cr2F(w) by optical measurrtriotits

is a difficiilt task. The data must be free of noise as two derivatives arc required. Howlvc~r.

t lit atialysis based on the optical data of &Cu0 has bec11 successful. [llarsiglio9S] Tlir

ostractcd n 2 F ( w ) by inversion of reflectance data structure is pro\.en to be in aclw~iiatr~

iigrecrncnt n-it h similar data obtained from neutron scattering results.

A niimber of theoretical interpretation have been offered to esplain the microsc-opic.

origiri of the pseudogap. These include nearly antiferromagnetic spin ff uctuations[C~iiiI~i1koi-91].

c-oritlcrisation of preformed pairs[Emery95h, Geshkenbein]. SO(5) symmetq-(Zhaiig971

Cliapter 2. SHEORY -4XD METHODS: IR Optical Properties of Solids :3(i

;iiicl spin-charge separation[Lee9T].Severai reviews esist[Randeria9i. Timusk98]. [Eni(>ry95hj

2.3 Optical properties of superconductors

-411 of tbc HTCS's eshibit a dramatic departure from the Drude form of the reflectivit!-.

I~istcacl of being close to 100% for frequencies below the plasma edge as especr(v1 for

a good rrictal, the reflectance of the cuprates drops gradually in a typical qi1,zci-liri(biir-

hshioii until 1 eV. Departure from the optimally cioped HTSC yields lower rcflectani-P.

Figiirc 2.5: The reai and irnaginary parts of the optical conductivity of a supercontfri(.tor. Tlic calciilation is based on an s-wave order parameter.

For a conventional superconductor, absorption cannot occirr until enough encrgy is

;~l)sorl~cti to break the Cooper pair. The threshold for absorption in an isotropic s - w i \ - c l

siipcrc~onductor is a t w = 2A. rather than A: because it is ncccssary to excite botIl t h c l

particles forming the Cooper pair abovc the gap energ'.. In Figure 2.5 we plot 01 ;uid (3-

for the siipcrconducting state of an isotropic s-wave superconductor. Since the pli>-sicxl

Chap ter 2. THEORY -4ND METHODS: IR O p tical Properties of Solids :3 1

tricariirig of ol is absorption then the real part of conductivity is identically zero 1 ) ~ h n .

L = ?A. Beyond this frequency the conductivity starts rising sharply until it reaciics t h

\.;due of normal s ta te conductivit. Note that the difference in spectral weight bct~vwri

tlio riormal (Fig. 2.4) and superconducting (Fig. 3.5) fi is shifted to the delta fiiri

C'Ii a p ter 2. THEORY =I!VD METHODS: In O p tical Properties of Solids

underdoped optimal

1 Tem perature

overdoped

Figiirc 2.6: Signature of the pseudogap in the temperature dependence of the iri-l)l;iiicl 1-osist,i\-ity tlic Hall effect coefficient and the susceptibility for tlic tlirce regions o f l i o f ( ~ c-oriccrit ration. (. Aftcr Batlogg et al.[Batlogg94])

C'liap ter S. THEORY -4ND METHODS: IR Optical Properties of Solids

(meVI O 20 40 60 80

- - - - - - - - - - - - - - - .......... .. ..,.

O 200 400 600 800 FREQUENCY (cm-')

Figure 2.7: The c-asis conductivity of a n undcrdoped Y123 crystal. (--1ftor Honies et ai. [Homes95]) As the temperature is lowered the pseudogap dcvolops. Tlio irisct: The NMR Itnight shift (normalized a t 300 I i ) is plotted as a functioii OC t cwipctrature for an underdoped Y123 crystal. The circles show the low frequency r-iisis c.a~itlrictivity for samples with the same doping level.

< J-

Chapter 2. THEORY AND LZ-IETHODS: IR Optical Properties of Solids :34

the characteristic temperature scale where the suppression occurs matches T' frorii spiri

siisccptibility measurements. The size of the gap along the c-axis could be estimatctl cli-

t lier from the energy position where the full suppression takes place or from the stiiït irig

poiiit of the suppression. -4s a result: two values were obtained for Y123: 200 cm-' aii(1

400 cm-'.

The cstended Drude mode1 has been widely used to describe the coherent rcspoiiscb

aloiig the ab-plane direction. There is clearly a correlation between the suppressiori i ~ i

t hc in-plane scattering rate and the pseudogap in the c-asis conductivity. This suggclsts

a siniilar meclianism. However. the energy scale of the pseudogap along the ab-pla~io is

twicc as large as that along c-axis.

Ir! the ab-plane the most dramatic manifestation of the pseudogap is obscrved iri t f i ~

scattering rate for

C'liapter 2. THEORY AND METHODS: IR Optical Properties of Soijds

O 1000 O 1000 O Io00 O 1000 Poo0

Wave Nurn ber (cm-')

Figure 2.8: Scattering rate and effective mass for underdoped cupratcs. (Aftcr Piichkov et al-[Puchkov96d]) The frequency dependent scattering rate. top row. i i ~ i d tlic niass renormalization. bottom row: for a series of underdoped cuprate siiprr(*ori- ductors. Thc scattering rate curves are essentially tcmperaturc independent ahow 700 mi-'. lmt dcvelop a depression a t low temperatures and low frequencies. The rff(w i w mass is cnhanced a t low temperatures and low frequencies.

Cliap ter S. THEORY AND METHODS: IR Op tical Properties of Solids :3G

smiples. The trends are (i) no suppression in the scattering rate is observed at T>T,

witli the exception of the slightly overdoped Bi2212 compound: and (ii) the T-dependcmcc

l~ecornes more pronounced at high frequencies. It should be noted that for the strorigly

O\-rrcloped Tl2201 sarnple the frequency dependence of l/r(w) is reminiscent of Fcrriii

liqiiirl bchavior, but the temperature dependence is still linear whereas a quadratic- des-

pcridencc is espected for a pure Fermi liquid.

For completeness the effective masses are shown below the scattering rates. r r ) / t r / =

pcaks at the frequencies mhere scattering rate is suppressed. The maiiimum vali~c. of

r r ~ / r n * varies between 1 and 4.

-411 of the previously measured materials represent a bilayer family of HTSC. Thcl 0x1

siriglc layw cuprate from the above study was TI2201 which mas avaliable as a strori::

01-crdoped sarnple.

Cliapter 2. THEORY -4ND AIETHODS: IR Optical Properties of Solids

Figure '2.9: Scattering rate and effective mass for optimally doped cuprates. ( Aftrr Puclikov et ai. [Puchkov96d]) The scattering rate has a degree of temperatiiro clcl- prrirlcrice at low frequeocics. In the superconducting statc the scattcring rate is deprrssrd a t low freclilencies.

C h p t e r 2. THEORY -AND METHODS: IR Optical Properties of Solids

Ad- 1

O Io00 O I o 0 0 O Io00 2000

Wave Number (cm-')

Figure 2.10: Scattering rate and effective mass for overdopcd cuprates. (After Puchkov et al[Puchkov96d]) In the overdoped sarnples the liigli-frequency s(ïit- twirig rate shows an increasingly strong temperature dependence. -4s part of tlir Iiigli- fictc~iicricy scattering disappears at low temperatures. t h e low-frequency depression of l / r (&) and the effective mass enhancement decrease in magnitude. This occurs c w r i iri t lie siipercoriducting state.

Chapter 3

THE PSEUDOGAP STATE OF UNDERDOPED La2-,Sr,Cu04

Tlir prescnce of a pseudogap in the normal state of the underdoped high tempcratiirc)

siipercontluctors is by nonr widely accepted.[Timusk98] The strongest eïidence for t l iv

pseiidogap state cornes from recent measurements of angle resolved photoemission spiv--

t ra [S larshall96a] as well as vacuum tunneling[TaoST. Renner98j. However. t hesr t w l i -

riiqiics both demand extremely high surface quality and have therefore mainly befw r(%-

st ricted to Bi2SrZCaC~208+6 (Bi2212) and YBa2Cu30ï-a (Y 123) materials. bot li n-i t li

two Cu02 Iayers per unit ceIl. Techniques that probe deeper into the sample siic.11 ;is

c l < - trarisport[Bucher93. Ito93. Batlogg94], optical conductivity [Basovgô. Puchl;o\.DGi-.

Piichkov9Ga] and S!vIR[Millis93] were not only the earliest to show evidence of tbr p w i i -

ciogap. but have been estended to a much larger varietj- of rnaterials. including sc>\-c.r;il

~iiaterials with one Cu02 layer.[Batlogg94] In al1 cases evidence for a pseiidogap lias Iwc~ii

rc~portrci.

Tlic pseudogap in LSCO as seen by NXIR and neutron scattering[.\Iason92] is rat l1c.r

wrak and Ilas led to the suggestion that the existence of the pseudogap in the spin rsi-ita-

t ion spcctruni is only possible in bilayer compounds such as 1'123 and YBa2Cu.10.1 ( \ - l ' ) A ) .

III particiilar. Alillis and Monien attributc the pscudogap (or ttic spin gap) to t tic st roilfi

i~ritifcrrornagnetic correlations between the planes in the bilayer. which are respoiisiblr

for ii quantum order-disorder transition. [Al illis931

.-\part from having only one CuOa layer. La2-,Sr,Cu0.1 (LSCO) is also a good tiiorlrl

systcni for the study of doping dependences since it can be doped by the addition of

Cliapter 3. THE PSE UD0G.W ST4TE OF CiNDERDOPED La2-,Sr, Cu04 -4 O

stroritiiini over a wide range: from the underdoped, where Tc increases with Sr curitt~ir.

to th^ optimally doped where Tc reaches its maximum value of x 40 K a t x = 0.17. a i c l

to the overdoped region where Tc --+ O at x = 0.34.[Batlogg94] T h characteristic signatures of the pseudogap state in the dc resisti\-it~[Bucli(~r9:3]

arcx sceii clearly in LSCO[Batlogg94. Uchida911. There are the st riking deviations I w l o ~ v

a tririperature T' from the high temperature linear resistivity. resulting in a clear I)i.(biik

i r i slopc at T a . It was found by B. Batlogg et al. [Batlogg94] that in LSCO Tg drcrws(~s

froiii 800 1< to approsimately 300 K as the doping level is increased from the stroiigly

iirictcrdoped to just over the optimal doped level. Similar behavior a t T = T' lias

I)rw observeci in the Hall effect coefficient and the magnetic suscepti bility. [H~mgD-l .

To pyg0961

The pseudogap can also be obsex-ved if the conductirity is measured in the frecpim.y

rloiiiairi. o(w). mhere it shows up as a striking depression in the frequcncy dcpcri

Chapter 3. THE PSEUDOGAP ST4TE OF LTVDERDOPED La2-,SrzCu04 41

the lpBa2Cu30i-, (Y123) and YBa2Cu40a (Y12-I) materials[Basov96, Homes93bj as ~ 1 1

as iri LSCO[Basov95a. C'chidaSû]. In slightly underdoped LSCO the pseudogap s t a t ~ i r i

tlic c-asis direction is not as well defined as it is in the two plane materials.[Basîsov9G1

However. as the doping is reduced further, the c-asis pseudogap state features below 0.1

r i * liecorne clearer. [Cichida961

Preïious work on the in-plane o(w) of the single Iayer lanthanum strontium ciipnitc

iiirlitclcs work on the osygen doped La2Cu04-s ~naterial[QuijadaSJ]~ tliin filuis of a

LSCO[Gno93] as well as work done on LSCO single crystal at room temperat ure[Cclii~Iit9 11.

To otir kriowledge: a study of the temperature and doping dependence lias not becri (loii(l.

\\c fil1 this gap here by performing optical measurements on high-quality LSCO siiiglv

crystals a t temperatures ranging from 10 K to 300 K at tmo different doping lc\-01s.

I)orli sliglitly underdoped. Also the optical properties of both the ab-plane and c-iisis of

Liii.nfiïn. inCuO.i were measured on the same crystal.

To better display the effect of increased coherence on 0 ( ; 2 ) . ~ resulting from tlio for-

rriatiori of the pseudogap state ive use the rnemory function? or estended Drude a~iiilysis.

111 this treatment the cornples optical conductivity is modeled by a Drude spectrurri \vit11

a frequencj--dependent scattering rate and an effective electron mass.[Got~ei2~ --Ul(m;l]

\\*hile the optical conductivity tends to empliasize Cree particle behavior, a study uf tlw

firqiirricy dependence of the effective scattering rate puts more 11-eight on displ-irig t l i r

irit cractions of the free particles with the eiemcntary excitations of the sp tem. [GoIcIs~!

Tt ic. tcmpcrature evolution of the frequency dependent scat tering ratc and effectiw tiiiiss

spcctra arc of particular interest and are defined as follows:

Cflapter 3. THE PSE UDOG.4 P STXTE OF UNDERDOPED La2-&-, CUO.* 42

1 00 200 300 Temperature, K

Figiirc 3.1: The temperature dependence of the in-pIane resistivity of Lal.8sSro.i.iC'i~0.i is sliown with a sharp superconducting transition at 36 K. Tlir slii111c of the curve is consistent with T* being greater than 300 K.

Chepter 3. THE PSEUDOG.4P ST4TE OF UNDERDOPED La2-,Sr,Cu04 4 3

H e m O(&. T) = 01 (w, T) + io2(u, T) is the complex optical conductivity and rr., is r iic plasina frequency of the charge carriers.

The single crystals of Laz_,Sr,Cu04: with approsimate dimensions 5 x 3 ~ 3 mm:'. n - ( v

grou-ri IIJ- the traveling-solvent floating zone technique at Oak Ridge [Cheng911 in t i i ~ ~ S P

o f -1. = 0.14 and in Tokyo [Kimura92] in the case of x = 0.13. The critical tcmprratiirr

was cletermined by both SQLID rnagnetization and resistivity measurements and \\-ils

fi~iinrl to be 36 I< for the nominal concentration of Sr x = 0.14 and 3'2 I i for .r = 0.l:J.

Siricc tiic highest Tc in the LSCO system has been found to be 40 I\: for x = 0.17. u-O

c.oiiclutle that both crystals are underdoped.

Tlic crystal with x = 0.14 was aligned using Laue diffraction and polisliecl parall(~1 to

t i i ~ CID.> planes. The crystal with x = 0.13 was polished in Tokyo to yield both ab-plmicl

aritl ac-plane faces. Both surfaces were measured. Polarizers were used for the a(.-fa(-r

( h t a to scparate the contribution of CuOl planes from the c-axis optical responsr.

To gct an uncontaminated ab-plane measurement it is important to have the sarii1)l~

siirfacc accurately parallel to the abplane to avoid any c - a i s contribution to tiic oprii.;il

ruiicliict ivi tu. [Orenstein88] Ttie rniscut angle between the polished surface normal aiicl t li(*

tu-asis was chccked by a high precision triple a i s s-ray diffractometer and was deterrriiiicvf

t o fw ICSS tiian 0.8%.

--!Il rcflcctivity measurements were performed witli a LIichelson intcrferomctcr iisirig

tlircc cliffcrerit detectors which cover frequencies ranging from 10 to 10000 cm-' (1.25 rr icx\ -

1-23 c l*) . The esperirnental uncertainty in thc reflectance data does not csccctl 1%.

Tlic dc rcsistivity measurements were carried out using a standard &probe tectiriiqiicl.

Tlic rcsiilt of the resistivity measurernent on the saine Lai .&ho.&~O.~ single c.r~-st al

iisod in tllc opt.ical nieasurements is shown in Fig. 3.1. It is commonly accepted tliat t l i c x

DC-rrsistivity is linear a t high temperatures for LSCO and tliat the pseudogap bcgiiis to

foriri iicar tiic ternperaturc where the resistivity drops below this linear trend.[Batiogp9-l]

Cliapter 3. THE PSEUDOG.4P ST.4TE OF UNDERDOPED La2-,Sr,Cu04

O SOO 1000 O SoO 1000

Wave Number, cm-'

Figure 3.2: The refiectance of L a l . s & r ~ . ~ . ~ C u O ~ (a) and Lai.&ro.l&uO.l (b). Tlir solid lines show the normal s ta te spectra. while the dashcd curve shows supw- (.orirliic:ting s ta te spectrum. The thinest line shows the spectrum a t the ternperatiir(. rlosrst to Tc. The insert in the leFt panel is a semi-log graph of the reflectancr ;it 300 I\; which shows a plasma edge around 7000 cm-'.

C'hpter 3. THE PSEUDOGAP ST-4TE OF UNDERDOPED La2-,Sr,Cu04

m *,

- - - - - fit lino

Figiiro 3.3: The real part of tlie dieIectric function as a function of d - 2 fol- Lai.sGSro.i-rCiiOcl a t 10 I i is shown in panel a) and for Lal.87Sro.inCu0.i at 25 II; is siion-ri in p a r d il). Ttic dashed line is linear fit. The slopc of the fit givcs the values of the Lonrioii pcrictration depth.

G'hpter 3. THE PSEUDOG.4P ST'ATE OF UNDERDOPED La2-,Sr,Cu04 46

.At l o w r tcrnperatures there is a region of superlinear temperature dependent rcsisti\-ity.

Tlir T' value for Our samples with x = 0.13 and x = 0.14 estracted from the phase dia-

grain of Batlogg et al.,[Batlogg94] are 650 K and 450 Ii, respectively. In agreement witli

tliis. tlic resistivity shows a superlinear temperature dependence below room temperat iirc

iis cspecteci in the pseudogap region.

111 Fig. 3.2 we present the reflectivity data a t temperatures above and belon- T, for

tlic two sampks. For clarity, only three temperatures are shon-n: T = 300 K. an intcrrricl-

dintc tcmperature above the superconducting transition and a lotv temperature (==: 10 11)

iii tlic siipcrconducting state. In the frequency region shown the reflectance is strorigl?-

tmiperature dependent for both rnaterials, dropping by approsimately 10% as thc t c m -

pcraturc is increaseci from the lowest temperature t o T = 300 K. The plasma dgct is

ol>scrveci at 7800 cm-' (see insert of Fig. 3.2). The distinct peaks a t approsimatcl?- 135

aiicl .365 cm-' in the LSCO reflectivity spectra correspond to the excitation of al>-pla~i

C'Oapter 3. T H E PSEUDOGAP S ï A T E OF UNDERDOPED La2-,Sr,Cu0.1 41

Frequency. cm-'

Figure 3.4: The reflectance of Lal.87Sro.iaCu0.1 with E I I c asis. Tlic temperature sequences are 10 K: 40 K: 150 K. 200 K. 300 I< and 400 K.

Cflapter 3. THE PSEU-DOGAP ST-4TE OF mDERDOPED La2-,Sr,Cu04 4s

J- = 0.13. For the Iiigh-frequency extension for u > 8000 cm-' ive used the reflcctirit>-

rcsiilts of Cchida et al.[Uchidagl] At frequencies higher than 40 eV the reflectivit~- \\.ils

assiinied to fa11 as l/w'!

11% calculatc the plasma frequency of the superconducting charge carriers and tlic

Loricloii penetration depth using the following formula:[Timusk89]

Tlic dope of the low-frequency dielectric function, el (w). plotted as a function of [le-'

i r i Fig. 3.3a.b gives plasma frequencies of 6100 cm-' and 5700 cm-' in the supercoiid~i(.t-

irig statc. The corresponding London penetration depths are AL = 1 / 2 ~ ~ + , ~ = 250 11111 a ~ i d

280 rim for Lal.8GSro.i4Cu04 and La1.87Sr0-13C~04. respectively. These values are in goo(1

iigrecrnent with those obtained previously by Gao et al. in filrns[Gao93] ( A L = 275i . i 1 1 nr )

iiiid IF muon-spin-relaxation[Aeppli8ï] ( A L = 250 nm).

T h c-asis reflectance of the x = 0.13 sample is shown in Fig. 3.4. The correspo~i(liiig

c.oridiictivity is low and is dominated by optical phonons (Fig. 3.5).

In 1-BCO 123 and 124 the pseudogap along the c - a i s manifests itself as a depressioii i11

rlir (-oridiictivity at low frequcncies.[Homes93b. Basa\-96. Homes951 Tlierc is no colimwt

Driidcl peak and the conductivity is flat and frecpency independent. In the tcrnpmiriirc~

aiid doping range where a pseudogap is espected a low frequericy depression o f I l i ( '

(writliictivity is scen with an cdge in the 300-400 cm-' region where the contlii(.t k i t ?

riscs to t lic high frequency plateau.

111 order to isolate the electronic features of our LSCO c-mis spectrum we rnagnify rli(1

low \-duc rcgion of olc (Fig. 3.5). There is no sharp pseudogap edge in the low-frr~l1i(~ri(->.

irifrarcd data for rindcrdoped LSCO as there is in the case of Y123. It is possiblc3 1 Iinr

siidi a fcature could be hidden under the large phonon structure. Efforts to siilm-a

Chapter 3. THE PSECiDOG.4P STATE OF CÏNDERDOPED La2-,Sr,Cu04 49

Frequency, cm"

Figrirc 3.5: The c - a i s conductivity of La1.8ïSr0.13C~04 is shown a t various temperatiirw Siricc tlic ptionon peaks are dominant in the direction perpendicular to C u 0 2 pla~ios. t h grapli is focused a t the background conductivity. The two inserts are the ( a - n i s coiidiictivity of the underdoped LaimiSro~laCu04 measured at 450 cm-' and a t GO0 cm-'. The c - a i s conductivity a t 430 cm-' is depresscd below 300 K. lio~vcvi~r. i t is alrriost constant above 600 cm-l. This could be a signature of the psciiciogali forrriation with the size of 500 cm-' for temperatures less than 300 K.

C'Ilapter 3. THE PSEUDOG.SP ST-STE OF UNDERDOPED La2-,Sr,Cu0.1 .5 0

sc!risitit-e to the choice of their shape in fitting procedures. Xonetheless. the ran' chta

clcarl~r shows that there is a low frequency depression of the c-auis conductivity. Coridiic--

t i \ - i t ~ - at 450 cm- ' is uniformly suppressed below T=300 K (Fig. 3.5insert), whcrras t l i i ~

wriductivity at 600 cm-' is nearly constant a t all temperatures. Based on this analysis

onr cari conclude that the pseudogap s ta te in the c-axis opens up below 300 K arid its

sizc is approsimately equal to 500 cm-'.

500 1 O00

W i v e Number. cm-'

Figiirc 3.6: TIic ab-planc conductivity of La1.87Sr0. 13C~0 .1 is shown at diffcrcnt tcmi 1 ) c b r - - at Ill-CS.

1Ianifcstations of the pseudogap in the ab-plane conductivity cxist as a loss of q w r t r d

Chapter 3. THE PSEUDOG.4P ST.4TE OF UNDERDOPED La2-,Sr,Cu04 5 1

w i y l i t between 700 and 200 cm-' balanceci increases both below and abovr rliis

frrcpcncy. In both Fig. 3.6 and Fig. 3.7 one can see the temperature evolution o f t l i ~

sliarp depression in ab-conductivity below 700 cm-' at temperatures above Tc. -4 iiiii 400 K. an order of magriitriclr Iiiglicr than the superconducting transition temperature Tc (32 K). This is significiii ir 1 -

cliff~rcnt frorn previous results on cuprates where T* more or less coincidcs with T,. rioiir

opt irrial doping.

The temperature dependence above 700 cm-' is strongly influenced by tlic Ic~vrl of

Sr tloping. in the undcrdoped sample the high frequency scattering rate is nearly t v i i i -

pc:riitiire independent up to a certain temperature (Fig. 3.8a and Fig. 3.9a). wiiicli inb

ivi1l cal1 T" abovc which a pronounced temperature dependencc of l/r(;.T) is swii

(Fig. 3.81, and Fig. 3.9b). In the x = 0.13 sample T** zz 200 I i while in the x = O. 14

siunpl(1 T" x 150 K. It is iteresting to note ttiat these values arc in a good agrc(.riicbiir

w i t l i tlic crossover temperature found in NMR expriment by \asuoka et al.[l'a~iiokiiD]

clcfiricd as the temperature where l /TIT has a maximum and cqual t o 175 I i ancl 1-10 I i

for .r = 0.13 and x = 0.14: respectively. We find that in the overdoped sampl(ls thcl

scat tcring rate a b o w 700 increases uniformly with temperature[Startseva9~t)] ;it

il11 tcniperatures siiggesting Te* + O in that limit. This behateior is also seen in otlicar

Chap ter 3. THE PSEUDOGAP ST4TE OF U.iVDERûOPED La2-,Sr, Cu04 .5 2

La,,,Sr ,-,, CU 0,rb plane f == 36 K

500 1 O00 Wave Number, cm"

Figiirc 3.7: The ab-plane conductivity of Lal-86Sro.i4Cu04 is shown at L-arious tcrnpbrii- turcs.

C'hapter 3 . THE PSEUDOGAP S ï X T E OF UNVERDOPED La2-&, -3 3

ovlc tliis rate is zz 1500 cm-'. Thesc scattcring rates are much highcr than wtiat is siwi

for tiic Iiiglier Tc materials reviewed by Puclikov et ai.[Puclikov96d] where at 3 0 0 I i

l / r o z 1000 f 200 cm-' for several families and man- doping levels. This liigh rwi~111al

scat tcring differentiates the LSCO material from the other cuprates.

If wc cal1 the frequency below which the scattering rate is suppressed the n6-pl:iric~

psclidogap uab = 700 cm-' ive find that it is clearly bigger than the c-asis pseuclogap f r ~ q i i ~ n q - d, z 500 cm-'.

Iii atldition to t he pseiidogap depth and the tempcraturc dependence. sel-eral ot1ic.r

fcxtiircs seen in Figures 9 and 10 should be mentioned. The position of the pseiitfogii~>

roriiaiiis at 700 cm-' for al1 temperatures. Tliere are also several peaks positioiircl a t

XO m i - ' in the scattering rate which complicate the analysis. particularly in t h (-;iw

o f t l i ~ sarriple with x = 0.14. These peaks bave been observed by other groiips aiid

h v r I~rrri iittributed to polaronic effects.[Yagil94. l] Another possible esplaiiatioii is

t l i o corrclation of the ab-plane conductivity with c-axis LO phonons. \\é ciid ot)st~r\-o ;i

cliffcrcncc in the contribution of LO phonons to tlie ab plane reflectance with diffiwmt

propagation directions. an effect first observed by Reedyk et al.,[Reed5?k921 ancl also swii

i i i t Iic k I I c vs. k 1 c spectra obtained by Tanner's group.[Quijada95] In Fig. 3.10 t liil

r

C'tiapter 3. THE PSEUDOGAP ST4TE OF L'IWERDOPED La2-ISrz Cu04 -5 4

1 . I ' l .

+ 8 ) - x = 0.1 3 ab plane . - -

- - - - 200 K - 400 K .

10 K $ = 300 K 1

O 1 O00 2000 3000 1 O00 2000

Wsve Number, cm-'

Figiirc 3.8: The frequency dependent effective scattering rate and the effective I tiass o f La1 .pÏSro. L3 Cu0.1- Top panel: the low temperature frequency dependent scattering ratc of' L a I .HÏS~O.lJC~O.l below T** (a) and above T" (b) is calculated using Equation ( 1 ) . Tlic orisct of the suppression in the conductivity corresponds t o a drastic cliangc i r i tlio frcqucncy dependence of the scattering rate below T*. ;\bave 700 cm-' ttic sca1- t rring rate is nearly temperature independent and has a linear frequency dcpenclmc-(1 lwlon. T". Below 700 cm-l the scattering rate varies as wi+6 and shows a stro~ig twiporatiirc dependence. Bottorn panel: The effective mass of Lal.s7Sro.i3Cii0.1 Iw- loiv T" (a) and above T"(b) is calculated using Equation ( 2 ) . The onset of t h

m. (Y -2 corresponds to the onset of the suppression in the scattering or i l i a~ i~c~~ien t of mc nitc.

Cliapter 3. THE PSEUDOGAP ST-ATE OF UNDERDOPED La2-,Sr,Cu0.i

couples io ab plane features can be seen in Fig. 3.11. A coniparisori between the pc- k. s i r i

tlir effective scattering rate at 450 cm-' and 580 cm-' to t he peaks in Im(- 1/c,) s1ion.s

t lir same strong correlation seen in many other cuprates.[Reedyk92]

For coinpleteness we also plot the effective mass of the underdoped sainples at Ion-

rn'( ) riscs to t(mperaturcs(Fig. 3 . 8 ~ ) and high temperatures(Fig. 3.9c.d). -4s espected. + a riiasimum of =v 3 forming a peak a t z 400 cm-'. This enhancement of the effwti\.cn

rn-(w) rriass iri tlie pseudogap state as well as the upper limit of 7 are similar to wliat lias

h c n pre~iously reported for Y123. Y124 and Bi2212.[Puchkov96d]

Brfore closing this chapter we compare our results with t h e da ta of Gao et al.[C;;ioD3]

011 La2-,Sr,Cu0.i+d films and Quijada et al.[Quijada95] on osygen doped La2CiiOi-,j.

Oiir rcsiilts in the underdoped case are comparable with those of the osygen (1opod

r~iatcrial. although Quijada et al. did not c a r y out a frequency dependent scat t~r i r ig

ratcl a~ialj-sis for their iinderdoped sample. The film results of Gao et al. arc quitc.

cliffcrrnt froin Our findings. The films used in that study Iiad a strontium le\-cl tliat

n-oiild correspond to optimal doping in crystals. However. t h e l / ~ ( d ) c u n w rI(v-iatt1

rriarkccllj? frorn what ive obsen-e for sIightly under and overdoped samples. The aritliors

porformed a n cstended Drude analysis and found a strongly temperatiirc dcpciidt~it

scattcririg rate e\-en a t low temperatures. This is in sharp contrast to our results n-liic.11

\vorild suggcst a verx weak temperature dependence in this temperature rcgion. Bascvl o r i

oiir work. tlieir samplcs should bc in tlie pseudogap statc since thcy have an x t--; i l i i (~ rich;ir

optimal cloping. Comparing these results with other systenis. in particular with T12202.

t IVO factors suggest the possibility that the films may be overdoped. First. their T,. \vas

t i c w 30 K. lower than that espected for optimal doping. Secondly. it is knowri ttiat

tt iv osygcn lcvcl in films can Vary substantially and in LSCO osygcn can have a iiitijor

iiifliieiicc ori t he doping level[ZhangH94]. On the other hand. w e cannot completrl~. i - i i l

Cliiipteï 3. THE PSEUDOG-4P ST-4TE OF KVDERDOPED La2-,Sr,Cu04

0.00 0.10 0.20 0.30 eV

0.10 0.20 0.30 4000 I ~ I ' I ~ i ' l ' l . - 1)

x = 0.14 ab plane -- b) 3500 - 3000 -

Wava Numkr, cm"

Figiirr1 3.9: Thc frequency dependent effective scat tering rate and the effective niiiss ~f L+.x&o. [., CuO.1. Top panel: thc high temperature effective scattering rate of Lal.8sSro.l.ICu0.1 belou- T" (a) and abovc T" (b) is calculatcd using Equation (1). -4bove 700 cm-' the scitt- tcring rate has a linear frequency dependence and is temperature independent bclow T" (a) and temperature dependent above T"(b). Below 700 cm-' thc scattcririg rato varies as w'+* and shows a strong temperature dependence. Bottom panel: Th

Chapter 3. THE PSE UDOGAP ST.4TE OF UNDERDOPED La2-LSr, Cu04 - - -> i

tliat the films better represent the bulk material. It is clearly important to measurc filrtis

u h r e the osygen content is controlled by selective annealing.

Ir1 conclusion. the optical da ta in the far-infrared region. taken on two underd

sarriple is not anc t ed by temperature up to certain temperature Tg*. This temperat ~ i r ( ~ is

oqiial to 200 I i in case of LaL87Sro.laCu04 and 150 K in case of Lai8GSro.i.rCuOl. AI>ovo

T" tlic liigli frequency scattering rate is temperature dependent. This behavior is i(l(m-

t ira1 to t lie Iiigh-frequency effective scattering rate of an overdoped HTSC. [Puchkoi-3fi(I ]

Oiir findings in the direction perpendicular to the C u 0 2 planes sliowed ttiat t h 8

rlq>rcssion of the c-axis conductivity is not as prominent as the one foiind in t h tir-O-

1;iyc.r HTSC. Xe\.erthcless. the signature of the pseudogap can bc! secn at freqii(hiic.ithi;

1)c~lon- 500 cm-' up to room temperature.

C h p t e r 3. THE PSEUDOGAP ST4TE O F UNDERDOPED Laz-,Sr, CUO.!

I .,-______ ac plane, E Il a, k ! c

I I I 1 200 400 600

Wave Number, cm-'

Figiirc 3.10: Comparison of the reflectance of Lal.s7Sro.lnCu04 nicasured from ah ~ ) l i i i i ( ~ wit 11 X- II c anci from ac plane with k L c.

Cliap ter 3. THE PSEUDOG.4P ST-4TE OF LTNDERDOPED Laz-,Sr, CuO, .j 9

5 0 0 1 0 0 1 5 0 0

Frequency, cm"

Fig~irc 3.11: Comparison of the position of the peak at 450 and S80 cm-' in the diclwt rice loss fiinctiori of the c-axis phonons with the room temperature effective scattcririg ra t c > of La , .H&. l.lCuO.l. Tlir corrrspondence of the peaks positions: width and thc relative strength suggrsts t h t tlir riaturc of the peak rnay lie in the coupling of a ab-plane spcctra to t h e c - a i s lorigit,iidinal optical phonons.

Chapter 4

THE PSEUDOGAP STATE OF OVERDOPED La2-,Sr,CuO 4

Tho pscudogap phenornenon in high temperature superconductors has been twI1 ~ s -

t;il)lislied with a nurnber of esperimental techniques. Probes such as SI1R.[\4%m-riiS9.

Tiiii~isk9S] angle resolved photoemission ( .ARPES)? [Slarshall96b] tunneling. [Tao8T. R(wiicrQS!.

drt-t ronic specific heat .[Loram94] IR spectroscopy.[TimiiskSS. Homes93b. PuchkovXjd.

Horiics%] and a number of transport measurements[Ito93. Batlogg94I al1 show evid(w[.

of a partial gap or a pseudogap in the normal state. This gap forms below a ternpcratriw

T g . n-liicli is well above the superconducting transition temperature Tc. -ARPES rrn-vils

t liat t h pseudogap has a d,?+ synmetry with the gap going to zero in the k, = k,

dirrctiori with a maxima at the zone boundary in the k, = k, = O direction. Trarisport

c~sprrirncrits stich as the DC and infrared conductil-ity show t hat tiic pseudogap fornis i r i

t h spcctriim of low-en erg^. electronic excitations that scatter the charge carricrs. -4s r 11v

psciidogap forms. the scattering rate. l / r ( w . T). is reduced for both the tcmperatiirc1 arid

s p ~ t ral rcgions where the gap occurs. Furthermore. infrared spectroscopy shows t l i i i t a

C'liapter 4. THE PSEUDOGAP ST,.ITE OF O\ERDOPED LaszSr, Cu04 G 1

of T*.[Lee97, Emery9Sb: Chubukov97] It is predicted that the two temperature scales art3

(>qua1 at optimal doping. Indeed. in most of the families of cuprates this is approsirriatdy

wliat is observed. In particular. optimally doped YBa2C~306.95(Y123) shows no sigri of n

psctidogap for temperatures 10 I i above Tc. In Bi2Sr2CaCu208+a(Bi2212)