Scanning Tunneling Microscopy Studies in both Real- and Momentum-space of theDoping Dependence of Cuprate Electronic Structure.

by

Kyle Patrick McElroy

B.A. (University of California at Berkeley) 1998

A dissertation submitted in partial satisfaction of therequirements for the degree of

Doctor of Philosophy

in

Physics

in the

GRADUATE DIVISIONof the

UNIVERSITY of CALIFORNIA at BERKELEY

Committee in charge:Professor J.C. Seamus Davis, Co-ChairProfessor Joseph Orenstein, Co-Chair

Professor Dung-Hai LeeProfessor Norman E. Phillips

Spring 2005

The dissertation of Kyle Patrick McElroy is approved:

Co-Chair Date

Co-Chair Date

Date

Date

University of California, Berkeley

Spring 2005

Scanning Tunneling Microscopy Studies in both Real- and Momentum-space of the

Doping Dependence of Cuprate Electronic Structure.

c© Copyright by Kyle Patrick McElroy 2005

All Rights Reserved

ABSTRACT

Scanning Tunneling Microscopy Studies in both Real- and Momentum-space of the

Doping Dependence of Cuprate Electronic Structure.

by

Kyle Patrick McElroy

Doctor of Philosophy in Physics

University of California, Berkeley

Professor J. C. Seamus Davis, Co-chair

Professor Joseph Orenstein, Co-chair

Several scanning tunneling microscope (STM) studies of the electronic struc-

ture of high Tc superconductors make up this thesis.

First high-resolution Fourier-transform scanning tunneling spectroscopy (FT-

STS) is introduced as as a new technique for study quasiparticles in the high Tc

superconductor Bi2Sr2CaCu208+δ . Using it a characteristic ‘octet’ of quasiparticle

states that determine the quasiparticle scattering processes was found. By analyz-

ing the wavevectors of quantum interference patterns generated by the scattering,

we determine the normal-state Fermi-surface and the momentum-dependence of the

superconducting energy gap |∆(~k )|. These are in excellent agreement with angle

resolved photoemission spectroscopy (ARPES). Another finding is the discovery of

very strong quasiparticle scattering at the Brillouin zone-face.

Next, doping dependence of nanoscale electronic structure in superconducting

Bi2Sr2CaCu208+δ is studied. At all dopings, the low energy density-of-states modu-

lations are analyzed according to the same simple model of quasiparticle interference

1

and found to be consistent with Fermi-arc superconductivity. The superconducting

coherence-peaks, ubiquitous in near-optimal tunneling spectra, are destroyed with

strong underdoping and a new spectral type appears. Exclusively in regions exhibit-

ing this new spectrum, we find local ‘checkerboard’ charge-order with wavevector

~Q = (±2π/4.5a0, 0) and (0,±2π/4.5a0)± 15%. Surprisingly, this order coexists har-

moniously with the the low energy quasi-particle states.

Finally, the dopant oxygen atoms are located and related to the various man-

ifestations of disorder in Bi2Sr2CaCu208+δ . In previous work a ‘1 eV feature’ was

predicted by angle resolved ultraviolet photoemission (ARUPS) to be associated with

local, non-hybridized, 2p orbitals of oxygen atoms. We detected this feature by spec-

troscopic mapping STM. It consists of a peak in the filled state tunneling conductance

at about −0.960 eV and is localized to ≈ 8 A. The number of these ‘-0.96V features’

are found to scale correctly with oxygen doping and they are therefore identified as

the dopant oxygen atoms. This feature is imaged with atomic-resolution along with

simultaneous low energy spectroscopic information on the cuprate electronic struc-

ture. Strong correlations between the oxygen map and the gapmap, (∆(~r) ), the

quasiparticle interference LDOS-modulations, and the low-bias topographic disorder

is found. Thus, for the first time, we can begin to understand how the dopant atoms

control the electronic structure of a cuprate high-Tc superconductor.

Professor J. C. Seamus Davis

Dissertation Committee Co-chair

Professor Joseph Orenstein

Dissertation Committee Co-chair

2

Acknowledgments

The list of people who have helped me get to this point is just to long to list completely.

So the short list follows. Thanks to my mother who taught me to ask questions and

gave me every opportunity. Thanks to Lee for tolerating my absence. I guess all

those late nights don’t look so bad after the 2 years of a 3,000 miles separation, huh?

Laura for keeping me honest and for listening to the hours of ranting while I blew off

steam. To Michele for years of talking right past each other as only McElroy’s can.

My little brother Colin who can finally beat me up and set me straight. Peter Leiser

Thanks to my advisor Seamus Davis. Your scientific guidance has been in-

valuable, as has you ability to put up with me. D.H. Lee for his help understanding

high school quantum mechanics.

To the physics department support staff Anne Takezawa, Donna Sakima, and

Claudia Trujilo. Without you I would have missed registration, not had a job some-

times, and failed out at least once. A shout out to Eleanore Crump the LeConte

building manager who kept the lights on.

Thanks to my labmate, gymmate, and former roomate Joan Hoffman for help-

ing me pull through a dark Ithacan winter. To my Ithaca post-doc Jinho Lee who lets

you rant and rant before putting you in your place. The Canadian wonder Christian

Lupien who really can answer any question.

Then the list without comment: Barry Barker, Sudeep Dutta, Jennifer Hoff-

man, Eric Hudson, Shuheng Pan., Doug Scalapino, Peter Hirschfeld, Sasha Balatsky,

Peete’s Coffee, American Steel, Smapty, Jacob Aldredge, Andrew Glasgow, Peter

Rusello, Mr. Harvey, Julie Walters, Tom Stoppard, Lara Lomac. This list represents

i

≈ 1% of the people directly responsible for the work within this thesis.

ii

For Dr. Carla McElroy

for giving me opportunities

iii

Contents

Abstract 1

Acknowledgments i

List of Figures vi

1 Background 1

1.1 Superconductivity Basics . . . . . . . . . . . . . . . . . . . . . . . . . 1

1.1.1 Phenomenology . . . . . . . . . . . . . . . . . . . . . . . . . . 1

1.1.2 BCS . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3

1.1.3 D-wave Superconductivity: dx2−y2 . . . . . . . . . . . . . . . . 5

1.2 Cuprates . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5

1.2.1 Materials Bi2Sr2CaCu208+δ . . . . . . . . . . . . . . . . . . . 7

1.2.2 Band Structure . . . . . . . . . . . . . . . . . . . . . . . . . . 9

2 Tunneling and STM Theory and Data 13

2.1 Tunneling Theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13

2.2 Data Taken . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16

2.2.1 Topography . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16

2.2.2 Spectra . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18

2.2.3 Spectroscopic mapping . . . . . . . . . . . . . . . . . . . . . . 19

2.2.4 FT-STS map . . . . . . . . . . . . . . . . . . . . . . . . . . . 19

2.2.5 work function . . . . . . . . . . . . . . . . . . . . . . . . . . . 22

iv

2.2.6 Asymmetry . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22

3 Characterizing a d-wave Superconductor with an STM 24

3.1 Quasiparticle scattering and STM . . . . . . . . . . . . . . . . . . . . 26

3.2 Cuprates and the “octet” model . . . . . . . . . . . . . . . . . . . . . 26

3.3 Data . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 27

3.3.1 A short digression on FOV and resolution . . . . . . . . . . . 27

3.3.2 Raw Data . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29

3.4 ~ks and ∆(θ) and internal consistency . . . . . . . . . . . . . . . . . . 33

3.5 E(π/a) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 38

4 Doping Dependence 41

4.1 Digression on Gapmap . . . . . . . . . . . . . . . . . . . . . . . . . . 41

4.2 FTSTS and Doping . . . . . . . . . . . . . . . . . . . . . . . . . . . . 51

4.2.1 Charge Order and Underdoped Bi2Sr2CaCu208+δ . . . . . . . 52

4.3 Implications and Conclusions . . . . . . . . . . . . . . . . . . . . . . 56

5 Sources of Disorder in Bi2Sr2CaCu208+δ 60

5.1 Gapmap and the Question Why . . . . . . . . . . . . . . . . . . . . . 60

5.2 Photoemission . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 61

5.3 Discovery . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 62

5.3.1 Spectral Characteristics . . . . . . . . . . . . . . . . . . . . . 62

5.3.2 Location of the Oxygen Atoms . . . . . . . . . . . . . . . . . 63

5.4 Correlation with Other Observables . . . . . . . . . . . . . . . . . . . 66

5.4.1 Definition of Terms . . . . . . . . . . . . . . . . . . . . . . . . 67

5.4.2 Results Sample ∆ = 45 meV . . . . . . . . . . . . . . . . . . . 67

5.5 Results ∆ = 55 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 70

5.6 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 71

v

A Atomic resolution and STM evaluation 72

A.1 On the Topic of Homogeneity . . . . . . . . . . . . . . . . . . . . . . 72

A.1.1 Homogeneity and Resolution . . . . . . . . . . . . . . . . . . . 73

A.1.2 Necessary Criterion . . . . . . . . . . . . . . . . . . . . . . . . 78

vi

List of Figures

1.1 Resistance of a Hg wire as a function of temperature . . . . . . . . . 2

1.2 The density of states for the BCS ground state . . . . . . . . . . . . . 4

1.3 The density of states of the BCS ground with a d-wave gap . . . . . . 6

1.4 A schematic phase diagram of the cuprates in the T vs. p plane . . . 8

1.5 The structure of Bi2Sr2CaCu208+δ . . . . . . . . . . . . . . . . . . . 10

1.6 The band structure of Bi2Sr2CaCu208+δ . . . . . . . . . . . . . . . . 12

2.1 Schematic diagram of tunneling and STM . . . . . . . . . . . . . . . 14

2.2 Typical topography of Bi2Sr2CaCu208+δ and of NbSe2 . . . . . . . . . 17

2.3 Typical conductance spectra of Bi2Sr2CaCu208+δ . . . . . . . . . . . 20

2.4 Typical conductance maps of Bi2Sr2CaCu208+δ . . . . . . . . . . . . 21

2.5 Typical Fourier transform conductance maps of Bi2Sr2CaCu208+δ . . 21

2.6 Typical Fourier transform conductance maps of Bi2Sr2CaCu208+δ . . 22

3.1 The schematic phase diagram of Bi2Sr2CaCu208+δ again . . . . . . . . 25

3.2 The expected wavevectors of quasiparticle interference patterns in a su-

perconductor with electronic band structure like that of Bi2Sr2CaCu208+δ

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 28

3.3 g(~r, V ) for 3 different energies in a single FOV . . . . . . . . . . . . . 30

3.4 g(~r, V ) for 3 different energies in a single FOV . . . . . . . . . . . . . 31

3.5 g(~r, V ) for 3 different energies in a single FOV . . . . . . . . . . . . . 32

3.6 Representative fits to FT-STS data and the dispersions of the different

wavevectors of the ‘octet’. . . . . . . . . . . . . . . . . . . . . . . . . 34

vii

3.7 The expected wavevectors of quasiparticle interference patterns in a su-

perconductor with electronic band structure like that of Bi2Sr2CaCu208+δ

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35

3.8 The electronic density of states modulations associated with antinodal

quasiparticles at energies near the gap maximum. . . . . . . . . . . . 40

4.1 ∆(~r) for five different hole-doping levels . . . . . . . . . . . . . . . . . 43

4.2 Examples of measured g(~q, E) for a variety of energies E as shown at

three doping levels. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 46

4.3 The “octet model” for five different hole-doping levels . . . . . . . . . 47

4.4 Masking far underdoped data. . . . . . . . . . . . . . . . . . . . . . . 50

4.5 Integrating masked data and the Γ map. . . . . . . . . . . . . . . . . 53

5.1 Spectroscopic and spatial properties of the -960 meV feature . . . . . 63

5.2 Oxygen maps and associated topographies . . . . . . . . . . . . . . . 64

5.3 ∆ versus twice the number density of the oxygen atoms . . . . . . . . 65

5.4 Histograms of the interoxygen spacing for two samples and a random

distribution. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 66

5.5 The data types to be correlated, oxygen map, High and low Voltage

topographs and gapmap . . . . . . . . . . . . . . . . . . . . . . . . . 68

5.6 Angular average of the normalized cross correlation for the combina-

tions of observables shown in figure 5.5. . . . . . . . . . . . . . . . . . 69

5.7 Angular average of the normalized cross correlation for the combina-

tions of observables shown in figure 5.5 but for the sample with ∆ = 55. 70

A.1 A topograph with a blunt tip which shows all the characteristics of

‘atomic resolution’ . . . . . . . . . . . . . . . . . . . . . . . . . . . . 73

A.2 A 3000A linecut with a blunt STM tip showing almost no gap variation 75

A.3 Multiple data sets on the same surface with different tips. . . . . . . . 76

viii

A.4 Three histograms of gap values taken on the same surface on consecu-

tive days with different TTGs . . . . . . . . . . . . . . . . . . . . . . 77

A.5 A large FOV gapmap and conductance map of an impurity state from

every surface used for this thesis. . . . . . . . . . . . . . . . . . . . . 79

ix

Chapter 1

Background

1.1 Superconductivity Basics

The electron-like excitations of a Fermi liquid or a superconductor are called quasi-

particles. They carry the same spin and magnitude of charge as an electron. Like a

free electron their energy, E, depends on their momentum, ~k, but not in the same

fashion: E = ~2k2/2m as for a free electron with mass m. In the many body state of

a solid the electrons interact strongly with the lattice, lattice vibrations (phonons),

and the other electrons.

In 1956 Landau[1] proposed a phenomenological theory of metals which places

the low lying excitations of a metal in one-to-one correspondence to those of a free

electron gas. The dominant effect of the many body interactions is to shift the mass

of the excitations from that of the free electron mass. In 1911 after being the first

person to liquify 4He, H. Kamerlingh Onnes set out to investigate the resistance of

metals at low temperature.

1.1.1 Phenomenology

The resistance of a metal characterizes the energy that is lost when you move charge

(electrons) through it. At higher temperatures electrons can scatter off of irregu-

1

Figure 1.1: Resistance of a Hg wire as a function of temperature (Ω vs. K). Thanks to

Andreas Engel for help in finding this figure[2].

larities in the crystalline structure. These defects can be nonperiodic atoms in the

structure, lattice vibrations (phonons), or even the other electrons. As the temper-

ature is lowered the number of phonons decreases, and the probability of scattering

off of another electron does also. This leads to a characteristic lowering of the re-

sistivity with temperature. When H. Kamerlingh Onnes[3] looked at metals as their

temperature is lowered he found something amazing. The resistance for some of them

suddenly dropped to zero at a finite temperature, Tc , Figure 1.1. It seemed that the

metal was entering a new resistance state, one Onnes called “superconductivity”. In

this state it seemed that electrons could travel across the metal without loosing any

energy.

Twenty-two years later, in 1933, two German researchers discovered another

property of this state. Meissner and Ochsenfeld[4] found that in addition to losing

resistance, the superconducting state was perfectly diamagnetic. All magnetic field

was expelled from the specimen even if it was cooled below Tc in the presence of such

a field.

Besides zero resistance and perfect diamagnetism, many other properties of

the superconducting state were discovered over the years. The dependence of Tc on

the average mass of the atomic lattice implied that phonons played an important role

2

in this new state[5, 6]. Also, the specific heat of Sn below Tc was found to have

the form ae−b/t[7]. Taken in concert these led to several insights into the supercon-

ducting ground state. The perfect diamagnetism led London to conjecture that the

superconductor is a“quantum structure on a macroscopic scale a kind of solidification

or condensation of the average momentum distribution”[8]. The exponential specific

heat seemed to imply the opening of a gap in the excitation spectrum of the electrons

in the superconducting state.

1.1.2 BCS

In 1950 Bardeen decided to try studying the problem anew. He and a post doctoral

researcher Leon Cooper found that a net attractive force between two electrons on

opposite sides of the Fermi surface, no matter how small, led to a bound state for

those electrons. By constructing a wavefunction of the correct combination of these

pairs they could stabilize a state with the gap in the density of states. J. R. Schreiffer

proposed a state that did exactly these things[9, 10, 11].

They proposed a trial, ground state wavefunction of the form,

|BCS〉 =∏

~k

(u~k + v~kc†~k↑c

†−~k↓)|0〉 (1.1)

with u~k and v~k are the variational parameters,c†~k↑ is the creation operator for an

electron of momentum ~k and spin ↑ and |0〉 is the vacuum state. In order for this

state to be normalized the condition |u~k|2 + |v~k|2 = 1 is needed. By minimizing this

trial wavefunction ground state energy it was found that

|u~k|2 =1

2(1 +

ε~kE~k

) |v~k|2 = 12(1 +

ε~kE~k

) (1.2)

|E~k|2 = |ε~k|2 + |∆|2 (1.3)

where ε~k is the electron dispersion above the transition, ∆ is the size of the gap that

opens to single particle excitations, and E~k is the quasiparticle dispersion below the

transition temperature.

3

∆

3-3 -1 1

Figure 1.2: The density of states in of the BCS ground state normalized to the normal

state density of states.

From the point of view of tunneling it is this last part that is most important.

One of the unique pieces of the BSC ground state, versus other superfluid states, is

this gap in the single particle excitation spectrum. It manifests itself directly in the

density of states, ns(E), as

ns(E)/nn(E) =

0 E < ∆

= dε/dE = E/(E2 −∆2) E ≥ ∆,

(1.4)

which looks like Figure 1.2.

Since (as I will describe in the next chapter) tunneling measures the density

of states and the density of states is directly related to ∆, information about the

condensed electrons can be seen easily in the tunneling current. The exact form of

the gap has been taken from tunneling and provided some of the best verifications of

the predictions of BCS theory[12].

4

1.1.3 D-wave Superconductivity: dx2−y2

It is widely believed that the superconducting state of the cuprates is described by

BCS theory with one important modification. In the simple theory presented above

the superconducting gap is a constant over the entire Fermi surface, and the paired

electrons have zero relative angular momentum: i.e. have an s-wave symmetry. In

the cuprates this is not the case.

In 1993 a strong anisotropy was found in ∆ along the Ferm surface [13]. The

variation is consistent with a d-wave symmetry of the wavefunction and later experi-

ments verified that the phase does indeed change sign around the Fermi surface[14, 15].

Modifying the BCS equations to account for d-wave pairing symmetry is

straightforward and was done by K. Maki[16]. It simply entails replacing the above

relations with ones that retain the wavevector dependence of the gap Equation 1.3

|E~k|2 = |ε~k|2 + |∆~k|2 (1.5)

Then, by adding up all the states at each energy one can see the expected

DOS for a d-wave superconductor. This is shown in Figure 1.3.

1.2 Cuprates

Before 1986 the highest transition temperature to superconductivity that had been

found in any material stood at 23 K (for Nb3Ge[17]). During this year Bednorz and

Muller reported superconductivity in the LaBaCuO ceramics at 30K when they were

doped with excess holes. This was quickly followed with increases in Tc to 90K in

YBa2Cu3O7−δ in 1987 and 93K in Bi2Sr2CaCu208+δ .

In 1987 P.W. Anderson[18] postulated out that the CuO2 planes were re-

sponsible for the insulating, metallic, and superconducting behaviors seen at various

dopings. These materials all have similar structures. They are quasi 2-dimensional:

made up of stacked planes with strong intraplane and relatively wear interplane cou-

5

∆

3-3 -1 1

2

1

0

Figure 1.3: The density of states of the BCS ground with a dx2−y2 gap state normalized

to the normal state density of states.

6

plings. The stoichiometric parent compounds have 1 electron per unit cell and thus

are predicted, by band structure, to be metallic. Instead, the hybridized Cu-O states

are split by Coulomb repulsion into two Hubbard bands separated by an energy scale,

U , smaller that other transition metal oxides (but still significant at around 2eV). The

superconducting state results from adding charge into this system and making the

electrons itinerant.

A schematic phase diagram of what is seen in experiments is shown in Figure

1.4. At dopings from about 0.6 holes per Cu to 0.26 these materials superconduct

with Tc ’s reaching a maximum near 0.16 holes per Cu. There are many additional

interesting phenomena in the cuprates. At lower doping near to the Mott insulating

phase as temperature is lowered a partial gap, or pseudogap, develops in the density

of states and the resistivity shows a dip. This seems to happen at a well developed

temperature T∗ above which the resistivity, ρ is not Fermi liquid like. In fact it is

linear in T. At high enough dopings, far enough from the Mott insulator, normal

Fermi liquid behavior is seen[19, 20, 21].

In our studies we are limited to materials with specific characteristics. First,

since STM is a surface probe we need to have well characterized surfaces. One of

the most reliable ways to achieve this is through cleaving the sample in situ along

one of it’s crystal planes. In the case of the cuprates their planar structure makes

this technique ideal. Another concern is that the electronic structure of the surface

resemble that the bulk crystal. Again the quasi 2-d nature of these materials makes

this more likely. As long at the cleave is not along a charged plane and the CuO2

planes are coupled weakly enough the 2-d surface should resemble the bulk. The

cuprate that best satisfies these criteria is Bi2Sr2CaCu208+δ .

1.2.1 Materials Bi2Sr2CaCu208+δ

The structure of Bi2Sr2CaCu208+δ shown in Figure 1.5 (a). The maximum Tc of this

compound is 96K[22]. As with all cuprate superconductors it contains CuO2 planes.

7

"Pseudogap"

Strange Metal

Metal

SuperconductorM

ott In

sula

tor

doping (p)

T

T*

X

Figure 1.4: A schematic phase diagram of the cuprates in the T vs. p plane adapted from

[19]. At zero doping the compounds are antiferromagnetic Mott insulators. At higher dop-

ings the electrons delocalize and become various conductors. These include, superconductor

at dopings between ≈ 0.6 to 0.26 and low temperature, a state characterized by a partial

gap or “pseudogap” at low dopings above Tc but below T∗, a strange metal characterized

by a resistivity that is linear in T unlike a Fermi liquid, and a Fermi liquid at high dopings.

8

The number of CuO2 planes per unit cell varies but for this compound it is 2. Between

these two layers is a Ca plane. Around these planes are SrO planes then BiO planes

in that order. To achieve superconductivity extra oxygen atoms are doped into the

crystal. These are believed to be located at interstitial sites in or near the BiO plane

(see Chapter 5). Each oxygen takes two electrons from the CuO plane. The BiO

planes bracket the unit cell ant therefore make for a neutral plane cleave; minimal

charge reorganization is expected.

The interplane distance between adjacent BiO planes is almost the same as if

they were coupled by van der Waals coupling only. While thin films do frequently

cleave between the CuO planes[23], bulk crystals almost always cleave between two

BiO planes. For these studies we will be reporting only on single crystals that all

cleaved between BiO planes.

Since all our data is taken on this plane some discussion of how it fits in with

the rest of the structure is needed. The filled 6p orbital that juts out of the Bi atoms

gives them a large corrugation (contrast) for an STM. Thus topographies of this plane

show the Bi atoms each of which is above a Cu atom in the CuO plane by about 5

A. There is one more feature of Bi2Sr2CaCu208+δ that is important to note. This is a

buckling of planes that runs along the b-axis of the crystal called the supermodulation.

This modulation relaxes strain caused by the mismatched lattice parameters in the

different planes and runs throughout the bulk of the crystal. Although the effect of

the supermodulation is strong in the BiO topographs it has a relatively weak effect

on the CuO planes and on bulk properties and so will be treated as a unimportant

for now.

1.2.2 Band Structure

In order to understand the electronic structure it is instructive to first take a small

digression about another solid state probe: angle resolve photoemission spectroscopy

(ARPES)[25]. In an ARPES experiment a beam of light is incident upon a surface.

9

BiO planea) b)

O3 above Sr Bi above O2

above Cu

O1

Figure 1.5: a) The idealized structure of Bi2Sr2CaCu208+δ . There are 2 CuO2 layers

separated by a Ca layer per unit cell. There are also 2 proximate BiO layers separated from

the CuO2 layers by SrO layers. Adapted from [24]. b) the same crystal viewed from above

looking down at the BiO plane. An important thing to note is that each Bi atom lies ≈ 5

A above a Cu atom in the CuO2 plane.

10

the absorbtion of the photons by electrons gives them enough energy to break free and

leave the surface. By analyzing the ejected electrons’ energy and momentum and using

conservation laws, the energy and momentum of the electrons before the emission

can be deduced. This allows for the mapping of the dispersions of quasiparticles in

momentum space.

A brief refresher of the basic results from ARPES in Bi2Sr2CaCu208+δ will

aide in understanding the results of the following chapters. When they are optimally

and overdoped the quasiparticles in the Bi and Tl based cuprates have a rather high

mobility. Their energy dispersions are well defined, have been mapped throughout

the first Brillouin zone, and are well matched by tight binding models. Figure 1.6

gives a contour plot of the E(~k) taken from a tight binding parametrization with

|c|E(~k) = t0 + t1(cos(kx)− cos(ky))/2 (1.6)

+t2(cos(kx) cos(ky)) + t3(cos(2kx) + cos(2kx))/2

+t4(cos(2kx) cos(kx) + cos(kx) cos(2kx))/2 + t5(cos(2kx) cos(2kx)).

The parameters were taken from[26] and represent ≈ %17 doping (t0 = 0.1305, t1 =

−0.5951, t2 = 0.1636, t3 = −0.0519, t4 = −0.1117, t5 = 0.0510). These contours are

called the contours of constant energy (CCE) for quasiparticles.

A few features stand out in this band structure. First, the quasiparticles are

hole like with their Fermi surface centered around the X point in k-space. Second,

a large flat band regions near the M gives rise to a van Hove singularity in the DOS

within a 100 meV of the Fermi energy (its distance from the Fermi surface depends on

doping and specific material properties). The final feature which plays an important

role in chapter 3, is the shape of the low energy contours of constant quasiparticle

energy (below ∆0)). They are shaped like bananas that are centered on the ΓX line

and curve out towards the M points.

11

X

M

Γ

X

M

a) b)

Figure 1.6: Contours of constant quasiparticle energy in the 1st Brillouin zone for the a)

normal state and b) superconducting state of Bi2Sr2CaCu208+δ . The Γ = (0, 0), M = (1, 0),

and X = (1, 1) points are labeled. In b) the gap function is shown schematically in the

center it is taken to be dx2−y2or ∆k = cos(kx)− cos(ky). The parameterization of the band

structure is taken from[26].

12

Chapter 2

Tunneling and STM Theory and

Data

2.1 Tunneling Theory

A schematic of tunneling is shown in Figure 2.1 b). When two metals, at different

potentials, are brought into close proximity electrons can flow between them. At

high potential differences, greater than the work function φ, this results in electrons

ripping from the potential wells of the metal ions and flying across the vacuum. But

quantum mechanics allows for electrons to “cross” the junction even for low potential

differences when the transport is classically forbidden. The process by which this

happens is tunneling.

Now by 1st order time-dependent perturbation theory (Fermi’s golden rule)

the current tunneling from metal 1 to the unoccupied states in metal 2 is given by

Ir = (2πe

~)|M1,2|2n2(E2)(1− f(E2)) (2.1)

where f(E) is the Fermi function (1− f(E2)) is the probability that the target state

is empty), M1,2 is the tunneling matrix element between the two states in metal 1

and 2, and n2(E2) is the density of states DOS in metal 2 at energy E2.

13

eVb

NsNn

EF

E

2∆

It

Vb

Figure 2.1: (a) A schematic diagram of an STM. The sample is biased relative to a sharp

tip that is close to the surface and the tunneling current is measured. (b) Normal metal

insulator Superconducting (NIS) tunneling.

To add up the total current flowing we integrate over the allowed states weigh-

ing by their probability of being occupied in metal 1 n1(E). This gives us

I1,2 ∝∫ +∞

−∞|M1,2|2n1(E)n2(E + eVb)f1(E)[1− f2(E + eVb)]dE. (2.2)

Finally the total current across the junction is the current traveling from 1 → 2

minus the current 2 ← 1.

I ∝∫ +∞

−∞|M1,2|2n1(E)n2(E + eVb)[f1(E)− f2(E + eVb)]dE (2.3)

To make this more useful a few simplifying assumptions will be made. First

we will assume that the low temperature limit is sufficient. Since all measurements

made in this work were taken at 4.2K, kB = 0.27 meV for all data. At these very

low temperatures (kBT ¿ E and kBT ¿ eVb) the Fermi function term looks like the

14

difference between two Heaviside step functions so

f1(E)− f2(E + eVb) =

0 E < 0

1 E1 < E < E2

0 E2 < E.

(2.4)

Another simplification is made possible by assuming that near the Fermi energy (low

Vb) the tunneling matrix element is independent of energy or k vector[11]. This leaves

a simplified expression for the tunneling current,

I = A|M |2∫ eVb

0

n1(E)n2(E + eVb)dE. (2.5)

One last simplification can be made if the tip’s (metal 1) density of states (DOS) is

featureless in the energies of interest. Experimentally, this is achieved by testing the

tip out on a sample with a known DOS.

Up until now the matrix element M1,2 has been swept under the rug. In

general it can be calculated using Bardeen’s formalism[27] for a particular junction

geometry and particular wavefunctions for metals 1 and 2. For a spherical tip and

planar surface (approximately the case for STM) this has been done[28] and gives

|M1,2|2 ∝ e2κR|ψ2(~r0)|2 (2.6)

with ψ2(~r0) being metal 2’s wavefunction at the center of metal 1’s hemispherical end

and κ = ~−1(2mφ), the minimum decay length for states into the barrier from tip

and sample with a workfunction of φ and electron mass m. Since the sample’s wave-

functions decay into the barrier with with |ψ2(~r0)|2 ∝ e−2κ(R+d) we get the expected

exponential dependence on tip-sample separation, s,

|M1,2|2 ∝ e(−2√

2mφs/~) ≈ e(√

φ(eV )s(A)). (2.7)

It is this exponential dependence that makes STM possible. Since there is

such a strong dependence of the current on s one can use negative feedback on the

15

tip’s location to hold it steady above a surface. In order to keep the STM tip stable

relative to the sample a bias voltage Vb is applied to the sample relative to the tip.

Then the separation between the tip and sample is found such that a certain set

current I tunnels between the two. Because of the exponential dependence of the

tip-sample current on their separation, z, negative feedback is able to hold the tip at

a constant distance. A simplified version of the tunneling equation is

I(V,~r, z) = I0e−z(~r)/z0

∫ eV

0

LDOS(E,~r)dE (2.8)

where I0 contains the matrix element, z0 contains the workfunction and is typically

≈ 1 A, and LDOS(E) is the local density of states. For the most part this is the

model for tunneling I will use and refer to as the tunneling equation

2.2 Data Taken

Although only tunneling current as a function of bias Voltage (Vb) and the location

of the tip in (x, y) = ~r and z are measured by STM, there are still a number of ways

this 4-dimensional data can be cut to yield information about a surface.

2.2.1 Topography

The first and most common STM data type is called the topography. By keeping the

bias Voltage constant, Vb and moving the tip across the surface in the x and y plane

(parallel to the plane of the surface) and measuring the z location at which the tip

current is a set value (called the set current, Is) we can measure the shape of the

surface with atomic resolution. Examples of a topographies on Bi2Sr2CaCu208+δ and

NbSe2 are shown in Figure 2.2 a) and b) respectively. It is important to note here

that the measured topography reflects the real surface location (or charge density)

only within the approximation that the integrated density of states does not depend

on position. NbSe2 is an example where this approximation breaks down. This

16

x 5

Figure 2.2: (a) Typical topography of Bi2Sr2CaCu208+δ on a 64nm FOV with Is = 100pA

and Vb = 100mV. The inset topography is zoomed in 5 times to show the individual atoms.

Each bright dot is a Bi atom, which is 5A above a Cu atom. Also, the supermodulation

is visible as a long wavelength oscillation in the vertical direction. b) NbSe2 surface over a

10nm FOV. The CDW is seen as a brightening of every third atom. This 3 unit cell feature

is entirely due to the second term discussed later with Equation 2.9.

17

material is in a charge density wave (CDW) state at low temperature. The CDW

state has a wavelength of 3 lattice constants and causes such large variations in the

charge density. In Figure 2.2 b) this can be easily seen as every 3rd atom in the

topograph appearing to stand up from the surface. By taking the difference between

the measured topography zm with∫

LDOS(E,~r)dE allowed to vary and an idealized

one where∫

LDOS(E,~r0)dE is kept a constant we see that

∆s = sm − si = z0 ln(

∫ eV

0LDOS(E,~r)dE∫ eV

0LDOS(E,~r0)dE

). (2.9)

For now this is treated as a relatively weak effect but will be discussed in more detail

in Chapter 5.

2.2.2 Spectra

By holding z constant and differentiating Equation 2.8 with respect to V we see that

LDOS(E = eV ) =ez/z0

eI0

dI

dV. (2.10)

This means that the local density of states is proportional to the differential tunneling

conductance, dI/dV or g(~r, V ). Since positioning the tip above the surface required

finding a z such that I(z, Vs) = Is this can be simplified to

LDOS(E = eV ) =1

Is

dI

edV. (2.11)

In order to minimize the noise in the bandwidth of the measurement we use a

lock-in technique to measure spectra. This technique exploits the particular depen-

dence of the current I(V ) on the different harmonics of an bias current modulation.

Taking the input signal to be Vb = V0 + Vmei(ωt+φ) and Taylor expanding the current

around V0 we get

I(Vb = V0 + Vmei(ωt+φ)) = I(V0) +dI

dV(V0)×Vmei(ωt+φ) +

d2I

2dV 2(V0)×V 2

me2i(ωt+φ) + . . .

(2.12)

18

By using the lock-in amplifier to measure the first harmonic we can measure g(~r, V )

with a much narrower bandwidth than the whole 3 kHz of the system.

Figure 2.3 shows two typical tunneling spectra on a Bi2Sr2CaCu208+δ surface.

Comparing it to Figure 1.3 it is clear that it has many things in common- the peaks

at the edge of the superconducting gap for example.

2.2.3 Spectroscopic mapping

Tunneling spectroscopy has been used for almost half a century to learn the bulk

properties of materials by measuring the DOS. By using a STM one can do better

than this. With a sharp tip the STM measures the local density of states (LDOS).

Any number of materials have varying electronic structure on these small scales.

By measuring the g(V ) all over a surface (like taking the topography) we can

generate a spectroscopic map. This 4 dimensional data set g(~r, V ) can’t be shown in

a simple Figure. Instead different cuts through it must be shown. Choosing a certain

bias Voltage, V ′ and showing g(~r, V = V ′) leaves the two independent variables of

surface position, x and y. In Figure 2.4 g(~r, V ) is shown versus x and y. This is called

a conductance map.

Barring any large variations in the integrated LDOS, a conductance map gives

us a measure of the LDOS(~r, E = eV ) =∑

k |ψ(r)|2δ(E − Ek). This means that it

is an image of the square of the electron wavefunctions of a surface. For the most

part we will assume that the integrated LDOS variations are small and will mention

when this approximation fails.

2.2.4 FT-STS map

To see periodic structures in g(~r, V ) we take its Fourier transform. The resulting

FT-STS map, g(~q, V ), is typically shown in units of 2π/a0 so the atomic lattice is at

(1, 0) and (0, 1). Some example g(~q, V ) are shown in Figure 2.5.

19

0.0-0.2 0.2-0.4 0.4-0.8 -0.6 0.6 0.8

0.0-0.2 0.2-0.1 0.1-0.3 0.3

Sample Bias (V)

Co

nd

uc

tan

ce

(a

rb)

a)

b)

Figure 2.3: Typical tunneling spectrum taken on the BiO surface of Bi2Sr2CaCu208+δ . a)

is taken from −0.8 V to 0.8 V with a feedback parameters of 800 mV and 800 pA. b) is

taken from −0.2 V to 0.2 V with a feedback parameters of 200 mV and 200 pA.

20

a) b)

-14mV 0mV

Figure 2.4: Conductance maps, g(V,~r), of a 64nm FOV in Bi2Sr2CaCu208+δ . The setup

current and Voltage are 100pA and 100mV respectively. a) In this map V = −14 mV. b)

In this map V = 0 mV.

-14mV 0mV

a) b)

(1,0)

(0,1)

(1,0)

(0,1)

Figure 2.5: Fourier transform conductance maps, g(V, ~q), of the data in Figure 2.4. The

atomic lattice positions at (1, 0) and (0, 1) are labeled. Peaks associated with the super-

modulation are indicated by the arrows.

21

Figure 2.6: Two I(z, V = V0). The black lines are fits to the spectra I = a + be−kz and

result in work functions of 3.8 eV and 3.6 eV.

2.2.5 work function

The last degree of freedom an STM operator has is the tip-sample distance, z. By

starting at the setpoint and moving the tip back one can measure I(z, V0) the current

as a function of position. Because z only comes into the tunneling Equation (2.8)

in the matrix element I0e−z/z0 , this allows us to measure z0 = ~/(2

√2mφ) on the

surface. Two typical I(z, V = V0) sets are plotted in Figure 2.6. We typically only

acquire data when the workfunction measures more than 3.5 eV.

2.2.6 Asymmetry

Besides the different cuts of the data through ~r, z, and V there is one more piece

of data I will talk about in chapter 5. This addresses the asymmetry seen in the

tunneling spectra in metal STM to cuprate tunneling. In general, the conductance of

metal-insulator-metal (MIM) and metal-insulator-superconductor (MIS) tunnel junc-

tion spectra does not show a great deal of positive-bias/negative-bias asymmetry. But

in STM on Bi2Sr2CaCu208+δ and NaxCa2−xCuO2Cl2.[29] the filled states (negative

sample bias) are much higher than the empty ones (positive sample bias). This has

been discussed in many theories[30, 31, 32] and may be related to eh proximity to

22

the Mott insulating phase. To quantify the degree of asymmetry and to see if it is

correlated with any other observables we define the ratio map R(~r) by

R(~r) =I(V )

I(−V )=

∫ eV

0dI/dV ′dV ′

∫ −V

0dI/dV ′dV ′

=

∫ eV

0LDOS(E)dE∫ −eV

0LDOS(E)dE

(2.13)

The last equality hold as long as the tunneling equation 2.8 does.

23

Chapter 3

Characterizing a d-wave

Superconductor with an STM

A complementary description of electronic structure to one ~r-space is one in mo-

mentum space (~k-space), accessible for cuprates via ARPES and optical techniques.

ARPES reveals that, at optimal p in the superconducting phase, the Fermi-surface

(FS) of hole-doped cuprates is gapped by an anisotropic energy gap ∆(~k) with four

nodes, and, below Tc, quasiparticles exist everywhere along the normal-state Fermi

surface[25, 33, 34].

Figure 3.1 is a the schematic phase diagram for Bi2Sr2CaCu208+δ again. For

the studies in this chapter we will concentrate on the region around the X in this

Figure, near the overdoped region. In the overdoped region of the phase diagram, since

the Mott insulator is far away, the electron states are Bloch wavefunctions and well

defined in momentum space. Since these electron states have translational invariance,

impurities or external perturbations are required to make them visible to a real space

probe like STM. This has been done in simple metals and semiconductors[35, 36,

37, 38]. What is expected when quasiparticles scatter in the cuprates is somewhat

different[39, 40] but results from the same processes.

24

"Pseudogap"

Strange Metal

Metal

Superconductor

Mo

tt Insu

lato

r

doping (p)

T

T*

X

Figure 3.1: The schematic phase diagram of Bi2Sr2CaCu208+δ again. The studies in this

chapter will were done near the X at slightly overdoping.

25

3.1 Quasiparticle scattering and STM

In a normal metal with no impurities or defects the density of states is uniform. In

this situation STM would not be the appropriate tool for learning about the system.

To make STM useful something is needed to break the translational symmetry. In the

following I will give a brief description of a simple model for the effects of impurities

on the LDOS.

We start with the unperturbed system with quasiparticle wave functions |~k〉.The LDOS is given by

LDOS(~r, E) =∑

~k,σ

|ψ~k(~r)|2δ(E~k − E). (3.1)

The addition of impurities, crystal imperfections, or any intrinsic electronic hetero-

geneity will then scatter quasiparticles elastically. This disorder potential V has the

effect (to first order) of mixing a quasiparticle state, |ψ~k,i〉, with another, |ψ~k,f〉, with

a probability according to Fermi’s golden rule of

Wi,f =2π

~|〈ψ~k,i|V |ψ~k,f〉|2nf (E). (3.2)

When these new mixed states are put into Equation 3.1 the resulting LDOS, has

modulations with a wavevector ~q = ~ki − ~kf and a probability that is proportional to

ni(E)nf (E), the joint density of states.

3.2 Cuprates and the “octet” model

Since Bi2Sr2CaCu208+δ is highly anisotropic, or quasi 2-dimensional, scattering effects

should be easy to see falling off like r−1 rather than r−2 as for 3-d materials. The

Fermi surface and quasiparticle dispersions were shown in Figure 1.6.

In all Bi2Sr2CaCu208+δ samples studied in detail there is a great deal of disor-

der of many types. These include vacancies in the Cu sites of the CuO plane present

in all samples[41], gapmap disorder[42, 43, 44, 45, 46, 47, 48, 49], and other types.

26

These represent scattering potentials of various strengths and with various length

scales. Their prevalence implies that quasiparticle states at all wavevectors would be

scattered.

The quasiparticle density of states at energy E, n(E), is proportional to

∫

E(k)=E

|∇~kE(~k)|−1d~k (3.3)

where the integral is performed over the CCE: E(~k) = E. Each ‘banana’ exhibits its

largest rate of increase with energy, |∇~kE(~k)|−1, near its two ends. Therefore from

Equation 3.3, the primary contributions to n(E) come from the octet of momentum-

space regions centered around the points ~kj; j=1,2,..8, at the ends of the ‘bananas’.

An example of such an octet is shown as red circles in Figure 3.2 a). A quasiparticle

located in momentum-space near one element of the octet is then highly likely to

be scattered to the vicinity of another element of the octet because of the large

density of final states there. Since any of these red circles can be the initial or final

state for scattering a total of 16 inequivalent wavevectors should be seen (since STM

measures the real LDOS and the Fourier transform of a real function is symmetric

about rotations of 180 only 1/2 of a g(~q, V ) data set is independent). Within this

model the locations of the ~ks are equivalent to the locations of the normal state Fermi

surface. This model has been developed in detail by D.-H. Lee et al.[39].

3.3 Data

3.3.1 A short digression on FOV and resolution

Before data is presented, a few notes are needed about what is needed to resolve the

effects discussed above. Firstly, wave-vectors in the first Brillouin zone, by definition,

correspond to wavelengths on the order of one to a few lattice constants. According

to Nyquist’s theorem, to resolve them one then needs at least sample at a resolution

of a two points for every lattice constant. This requires an STM tip with good spatial

27

a)

b)

kx (2π/a)

ky (

2π/a

)

X=(1/2,1/2)M=(0,1/2)

q1

q2q3

q4q5

q6

q7

-1 -1/2 0 1/2 10

1/2

1

q1

q2

q3

q4q5

q6

q7

qx (2π/a)

qy (

2π/a

)

Figure 3.2: The expected wavevectors of quasiparticle interference patterns in a supercon-

ductor with electronic band structure like that of Bi2Sr2CaCu208+δ . a) Solid lines indicate

the k-space locations of several banana-shaped quasiparticle CCE as they increase in size

with increasing energy. As an example, at a specific energy, the octet of regions of high

|∇kE(k)|−1 are shown as red circles. The seven primary scattering ~q-vectors interconnecting

elements of the octet are shown in blue. b) Each individual scattering ~q-vector from this

set of seven is shown as a blue arrow originating from the origin in ~q-space and ending at a

point given by a blue circle. The end points of all other inequivalent ~q-vectors of the octet

model (as determined by mirroring each of the original seven in the symmetry planes of

the Brillouin zone) are shown as solid green circles. Thus, if the quasiparticle interference

model is correct, there would be sixteen inequivalent local maxima in the inequivalent half

of ~q-space detectable by FT-STS.

28

resolution (see Appendix A and a data point every angstrom or two.

Secondly, because E = ~v⊥,∆|~k|, the dispersions of these modulations, ∂E/∂k,

are on the order of the velocities around the fermi surface, v⊥/~ and v∆/~. In

Bi2Sr2CaCu208+δ these velocities have been measured by ARPES and found to be

on order 105 and 5 × 103 m/s respectively. These values correspond to a 1 to 0.05

AeV near the node. But, v∆ is a strong function of location on the Fermi surface:

going from its maximum near the node to nearly zero at the zone face. To resolve

dispersions of this size one needs a large FOV. The resolution of a discrete Fourier

transform is set by the size, l, of the data set in question: resolution ≈ 2π/l. So for

good enough resolution to resolve the dispersions expected one needs a FOV that is

at least 200a0.

The last consideration is the intensity of these modulations. Because the whole

framework described requires states that are reasonably well defined in momentum

space the intensity of scattering cannot be too great. If the modulation intensity were

100% of the DOS this would imply localized states and infinite width in momentum

space. A cautious guess would be that the modulations would only compose around

5% of the total LDOS. Since the noise floor of our STM is around 4 pA to 300 kHz

or 100 fA/√

Hz, and for stability we normally operate at 109 Ohm junction resistance

at 100pA we need to average around 10ms to get a S/N of about 10.

Putting all of this together, 2 points/a0, 200a0 in the FOV, and 10mS/ point

makes for 30 minutes per energy you want to measure. Now in practice to establish

feedback above the surface and to allow for the lockin amplifier’s rise time this time

is about doubled, leaving us with the ability to take the optimized FT-LDOS data

set for a full range of energies ±90 meV in 3 days.

3.3.2 Raw Data

Atomic resolution images of the LDOS and the resulting Fourier-space[50] images of

the wavevectors making up the LDOS modulations are shown in Figures 3.3, 3.4, 3.5.

29

Figure 3.3: g(~r, V ) for 3 different energies in a single FOV

30

Figure 3.4: g(~r, V ) for 3 different energies in a single FOV

31

Figure 3.5: g(~r, V ) for 3 different energies in a single FOV

32

Representative examples of the real space g(~r, V ) at each energy is shown in the first

column. All g(~r, V ) were acquired using the same atomic resolution and register. The

second column shows the corresponding g(~q, V ) attained by Fourier transforming the

g(~r, V ) adjacent to it. The only non-dispersive signals are due to the supermodulation.

The dispersion and evolution of all the wavevectors of these modulations is evident

in the differences between frames. Careful examination reveals that, in addition to

the slowly dispersing q1 signal moving to |q| < π/2a0 with increasing energy, there

is no additional signal above the noise at ~q = (1/4, 0)2π/a0 or ~q = (0, 1/4)2π/a0 (as

proposed for a coexisting charge density wave (CDW) order parameter)[51, 52]. The

agreement between these data and the simple prediction in Figure 3.2 b) is striking.

All 16 inequivalent wavevectors are present and evolve as predicted[53].

3.4 ~ks and ∆(θ) and internal consistency

The g(~q, V ) data can be further analyzed to check its quantitative agreement with

the theory. Between −6 mV and −30 mV locations of about 50 different q vectors

can be measured. To begin with we assign to each their ~qi designations. We then fit

to each a Lorentzian peak function (and example of various fits at 1 energy are shown

in Figure 3.6 a)) to measure the magnitude of |qi| and plot them in Figure 3.6 b).

Notably, |q4| is missing from this Figure. The reason for this omission is the peculiar

shape of ~q4 in the data. Because of its strange shape, assigning a center and therefore

a length unambiguously is not possible.

From these data and the “octet” model, the normal state Fermi surface lo-

cation, ~ks, and energy-gap, ∆k, can be measured because each |qi| is related to

33

0 5 10 15 20 25 30 35 40 0.0

0.4

0.8

1.2

1.6

2.0

q1

q5

q7

q2,6

q3

0.0 0.4 0.8 1.2 1.6 2.0 2.4 0

1

2

3

90o

115o

135o

7q 3

q

5q

5q

1q

1q

2q

6q

a) b)

Figure 3.6: a) Representative linecuts through the FT-STS data at the indicated angles.

These were taken from the FT-STS data in Figures 3.3, 3.4, and 3.5 at -12 meV shown

in the inset. The different q-vectors are identified with the ones with their approximate

position in the ‘octet’ model. b) The dispersions of each of the q-vectors int the data shown

in Figures 3.3, 3.4, 3.5. The widths of the peaks are shown on the last points.

34

0 10 20 30 40 50 60 70 80 900

5

10

15

20

25

30

35

40a)b)

0.0

kx (2π/a)

ky (

2π/a

)

θk

∆(θ

) (

me

V)

0.1 0.2 0.3 0.4 0.50.0

0.1

0.2

0.3

0.4

0.5

FT-STS (-)

FT-STS(+)

ARPESFT-STS (-)

Figure 3.7: a), The locus of scattering ~ks extracted using only the measured position of

scattering vectors q1 to q7 (excluding q4). The solid line is a fit to the data, assuming

the Fermi surface is the combination of a circular arc joined with two straight lines. The

grey band represents measurements of the Fermi surface location made using the ARPES

technique[54]. The error bars represent the statistical variations in a given ~ks when it is

calculated using at least five different |qi|. b), A plot of the energy gap ∆(~ks) determined

from the filled-state measurements, shown as open circles. These were extracted using

the measured position of scattering vectors q1 through q7 (excluding q4) in 3.6 b). The

solid line is a fit to the data. The filled circles represent ∆(~ks) determined using ARPES

techniques[54]. The red, open triangles are the ∆(~ks) as determined from the unoccupied

state measurements at positive bias as described in the text. They are in very good agree-

ment with those from the filled states. The mean value of ∆0 for this near-optimal sample

was 39 meV.

35

~ks = (kx, ky) by

~q1 = (2kx, 0) (3.4)

~q2 = (kx + ky, ky − kx) (3.5)

~q3 = (kx + ky, ky + kx) (3.6)

~q4 = (2kx, 2ky) (3.7)

~q5 = (0, 2ky) (3.8)

~q6 = (kx − ky, ky + kx) (3.9)

~q7 = (kx − ky, ky − kx). (3.10)

Because of crystal symmetry ~q2 is degenerate with ~q6 and so the two are not unique

leaving 6 different realtions. With Equations 3.4-3.10 we have a greatly overdeter-

mined set of equations for ~ks = (kx, ky). In fact for both kx and ky there are 20

different ways to extract it from the data. Inverting the data from Figure 3.6 b). For

each ~ks we used at least 5 independent combinations of ~q’s, and the average of all

such combinations yields our best knowledge of ~ks.

For −6 mV > V > −30mV, this is shown as open circles in Figure 3.6 b). We

interpret it as k-space trajectory of the ends of the CCE ‘bananas’ or, in interference

models[39, 53, 50], the normal-state Fermi surface. Our best fit Fermi-surface is then

the solid line 3.6 b). The ‘error’-bars surrounding the data points are the standard

deviation in each resulting from the different combinations of used and their small

size indicates how consistent all the ~q’s are with each other and the Fermi-surface.

We can also measure the superconducting energy gap function ∆(~ks) from

the data in 3.6 b). For a given energy, E = eV , we determine the associated ~ks

and plot E(~ks), the energy necessary to create quasiparticles along the trajectory ~ks.

Within our model this is the momentum dependence of the superconducting energy

gap ∆(~ks). Following ARPES notation, ~ks is parameterized by using the angle θk

about (π, π). Our results for ∆(θk) are shown as open circles in Figure ?? b). A fit

36

to the data is shown by the black line and is given by

∆(θk) = ∆0[(1−B) cos(2θk) + B cos(6θk)] (3.11)

with ∆0 = 39.3 meV and B = 0.182.

It is now appropriate to discuss the various implications of these results. First,

to compare ~ks and ∆(~ks) from FT-STS with those from photoemission, we plot the

Fermi surface estimated from ARPES (on a sample of similar doping[54]) as the grey

band in Figure ?? a) and the ARPES-derived ∆(~ks) as solid circles in ?? b). We

note the good agreement between these results from two very different spectroscopic

techniques. This agreement demonstrates the link between ~r-space and ~k-space char-

acteristics of the copper oxide electronic structure. It gives enhanced confidence in

both techniques because the matrix elements for photoemission and tunnelling are

quite different. Additionally, it demonstrates that the proposed attribution of the

nanoscale electronic disorder detected by STS[42, 43, 44, 45, 46, 47, 48, 49] to surface

damage not present in ARPES studies cannot be correct.

Second, the high-precision g(~q, V ) data presented here are relevant to pro-

posals that LDOS modulations in Bi2Sr2CaCu208+δ might result from the existence

of a second charge-density wave order parameter with fixed ~q -vector, for example,

stripes[51]. In such models g(~q, V ) should exhibit only two non-dispersive peaks (or

four non-dispersive peaks for twinned stripe domains). By contrast, quantum inter-

ference models predict 16 sets of dispersive ~q -vectors consistent with each other, the

Fermi-surface, and ∆(~ks). Clearly the latter is far more consistent with our data.

More recent proposals[52, 51] suggest that a set of non-dispersive LDOS modula-

tions due to fluctuations of a charge- or spin-ordered state might coexist with the

quasiparticle interference patterns. We do not observe such a non-dispersive signal

in addition to the quasiparticle interference effects in the g(~q, V ) of these as-grown

samples (except for crystalline effects). In principle, however, one cannot rule out the

possibility of such a hypothetical non-dispersive signal because its intensity could be

arbitrarily weak. Overall, our data demonstrate that quasiparticle interference is by

37

far the predominant effect.

Third, we discuss the quantum mechanical description of the copper oxide qua-

siparticles. Bogoliubov quasiparticles are the excited states of a conventional (BCS)

superconductor: quantum-coherent mixtures of particles and holes. It is important

to determine whether the copper oxide quasiparticles are of this type. A strong ex-

perimental indication consistent with Bogoliubov quasiparticles is the presence of two

identical branches of quasiparticle dispersion E(~k).By using FT-STS we can probe

the momentum-space structure of the positive branch by measuring g(~q, V ) at posi-

tive sample bias (tunneling into unoccupied states). If identical positive and negative

branches E(~k) exist, the LDOS modulations at positive bias should be consistent

with the negative-bias ∆(k) in Figure 3.7 b). When the positive branch interference

wavevectors g(~q, V ) are measured, the deduced ∆(k) (triangles in Figure 3.7 b)) is in-

distinguishable within errors from that of the filled-state ∆(k) (open circles in Figure

3.7 b). This provides evidence that, in momentum-space, the copper oxide quasipar-

ticles are particlehole superpositions, consistent with the Bogoliubov description.

3.5 E(π/a)

A final new FT-STS observation relates to the antinodal quasiparticles which are at

the heart of high-Tc superconductivity. Measurements of g(~r, V ) reveal intense LDOS

modulations with wavevectors equal to the reciprocal lattice vectors G when E ≈ ∆0

or equivalently when ~k ≈ (π/a0, 0). In a crystal, the electronic wavefunctions are a

linear combination of states with wavevectors ~k and ~k + ~G near the zone boundary.

This mixing is due to Umklapp scattering off the crystal lattice and can produce

intense LDOS modulations at ~G when ~k ≈ (π/a0, 0). However, as shown in figure

3.8, we unexpectedly find that for a given E the Umklapp LDOS modulation signal is

localized to the nanoscale regions where E is equal to the local gap value. This implies

strong nanoscale spatial variations in the quasiparticle dispersions near ~k = (π/a0, 0)

and therefore significant scattering. Thus, whatever the source of nanoscale electronic

38

disorder, it appears to strongly influence the lifetimes of antinodal quasiparticles in

Bi2Sr2CaCu208+δ .

39

-24 mV

-50 mV-34 mV

40Å

d)

a)

20 mV

60 mVb)

c)

Figure 3.8: The electronic density of states modulations associated with antinodal quasi-

particles at energies near the gap maximum. a) A map of the energy-gap magnitude ∆(~r)

in a particular area of the surface studied in this paper (see color scale). b)→d), g(~r, V )

measured at three energies, -24 meV, -34 meV and -50 meV respectively, in this exact field

of view. One can immediately see, by comparison of panel a with the others, that wherever

E equals the local value of ∆, an intense ‘tweed’-like pattern exists g(~r, V ) . The wavevec-

tors of this pattern are the same in all three panels, either ~q = (2π/a0, 0) or ~q = (0, 2π/a0).

Thus, LDOS modulations consistent with Umklapp scattering occur at different energies in

adjacent nanoscale regions, signifying strong scattering of the antinodal quasiparticles.

40

Chapter 4

Doping Dependence

At a fixed low temperature, those quasi-particle states with ~k = (±π/a0, 0) and

(0,±π/a0), near the 1st Brillouin zone-face, degrade rapidly in coherence with re-

duced doping until they have become incoherent at p < 0.10[25, 33, 55, 56]. This is

a mysterious phenomenon which is closely correlated with superfluid density[57] but,

within conventional superconductivity theories, should not be related to it. By con-

trast, states on the ‘Fermi-arc’ (FA)[58] near the nodes retain their coherence down to

the lowest dopings studied[25, 59, 60]. Transient grating optical spectroscopy studies

of non-equilibrium quasi-particles in underdoped YBa2Cu3O6.5[61] also find lifetimes

for antinodal excitations that are orders of magnitude shorter than those of the nodal

quasi-particles. Thus, the electronic structure of underdoped cuprates also appears

to be heterogeneous in ~k-space, in the sense that states proximate to the gap-nodes

(nodal) have quite different characteristics and evolution with doping, than those near

the zone-face (antinodal). No generally accepted explanation exists for either these

~k-space phenomena or their relationship to the phase diagram.

4.1 Digression on Gapmap

Applying all the techniques from Chapter 2 in concert, we study the doping-dependence

of electronic structure in a series of Bi-2212 samples. They are all single crystals grown

41

by the floating zone method. Doping is controlled by oxygen depletion so that no

other elemental impurities are introduced. Each is cleaved in cryogenic ultra-high

vacuum before immediate insertion in to the STM head. If its BiO surface is flat and

free of nanoscale debris, each sample is usually studied for several months, typically

in a 50nm square FOV.

A serious complication for experimental exploration of these ideas is the fact

that real-space (~r-space) probes generally detect nanoscale spatial heterogeneity in

the electronic/magnetic structure of underdoped cuprates. For example, when 0.03 <

p < 0.14, muon spin rotation (µSR) studies indicate the existence of a disordered mag-

netic ‘spin glass’ in La2−xSrxCuO4 (La-214) and oxygen doped LaCuO4+δ,[62] and in

Bi2Sr2CaCu208+δ (Bi-2212).[63] STM based LDOS imaging reveals nanoscale elec-

tronic structure variations in Bi2Sr2CaCu208+δ [46, 48, 45, 49, 47] and in

NaxCa2−xCuO2Cl2.[64] Nuclear magnetic resonance (NMR) points to strong nanoscale

carrier density disorder with variations in local p of at least 25% of mean carrier den-

sity in both underdoped La-214[65, 66] and underdoped Bi-2212.[67] For a wide variety

of underdoped cuprates, scaling analyses of penetration depth measurements reveal

finite size effects consistent with nanoscale heterogeneity in the superfluid density.[68]

These results provide abundant independent evidence that spin and charge degrees of

freedom are heterogeneous at the nanoscale in many underdoped cuprates. It has not

yet been possible to determine if this heterogeneity is a sample-specific and extrin-

sic effect due to crystal, dopant, or chemical disorder, or is an intrinsic effect of the

cuprate electronic structure. Nor have its implications for the phase diagram been

considered widely.

Figure 4.1 we show 50nm-square gapmaps measured on samples with five dif-

ferent dopings. Identical color scales representing 20 meV< ∆(~r) < 70 meV are

used for all images. The local hole concentration is impossible to determine di-

rectly, but we estimate that the bulk dopings were approximately 4.1A(0.19± 0.01),

4.1B(0.18± 0.01), 4.1C(0.15± 0.01), 4.1D(0.13± 0.01), 4.1E(0.11± 0.01). Near op-

42

-100 50 100

20 mV

70 mVa)

b)

c)

d)

e)

f)

50 nm

11

2 3 4

5 6

Sample Bias (mV)

dI/

dV

(n

S)

1.0

0.8

0.6

0.4

0.2

0.0-50 0

Figure 4.1: (a)-(e) Measured ∆(~r) for five different hole-doping levels. f) The average

spectrum associated with each gap value in a given FOV. They were extracted from the

g(~r, E) that yielded Figure 3C but the equivalent analysis for g(~r,E) at all dopings yields

results which are indistinguishable. The coherence peaks can be detected in #’s 1-4.

43

timal doping (Fig 4.1A,B) the gapmaps are heterogeneous but nonetheless the vast

majority of tunneling spectra are manifestly those of a superconductor (see below).

However, at the lowest dopings and for gap values exceeding approximately 65 meV,

there are very many spectra where ∆ actually becomes ill defined because coherence

peaks do not exist at the gap edge (see for example Figure 4.1F, spectrum 6). We

represent these spectra by black in the gapmap, since they are almost identical to

each other and appear to be the limiting class of spectra at our lowest dopings.

The spatially averaged value of ∆(~r) for each crystal, ∆, and its full width

at half maximum, σ, are: 4.1A(∆ = 33 ± 1 meV, σ=7 meV), 4.1B(∆ = 36 ± 1

meV, σ=8 meV), 4.1C(∆ = 43 ± 1 meV, σ=9 meV), 4.1D( 48 ± 1 meV, σ=10

meV), and 4.1B(∆ > 62 meV but with σ ill defined). As doping is reduced, ∆

grows steadily consistent with other spectroscopic techniques, such as ARPES and

break-junction tunneling,[25, 33, 69] which average over the heterogeneous nanoscale

phenomena. This observation is very important because it demonstrates that our

Bi-2212 surfaces evolve with doping in an electronically equivalent fashion to those

studied by the other techniques, and that we are probing the low temperature state

of the underdoped pseudogap regime.

In Figure 4.1F we show a series of the ’gap-averaged’ spectra. Each is the

average spectrum of all regions exhibiting a given local gap value (from the single

50nm FOV of Figure 4.1C). They are color-coded so that each gap-averaged spectrum

can be associated with regions of the same color in all gapmaps (Figure ’s 4.1A-

E). The spectra are labeled from 1-6, with numbers 1 through 4 providing clear

examples of what we refer to as coherence peaks at the gap edge (indicated by the

arrows). These gap-averaged spectra are consistent with data reported previously

by from gapmap studies by Matsuda et al.[47] Here, from our doping dependence

study, we can report that this set of gap-averaged spectra is almost identical for all

dopings. The dramatic changes with doping seen in ∆(~r) (Figure 4.1) occur because

the probability of observing a given type of spectrum in Figure 4.1F evolves rapidly

44

with doping. For example, the gap-averaged spectrum labeled as 1 in Figure 4.1F has

a 30% probability of occurring in gapmap 4.1A, 25% in 4.1B, 5% in 4.1C, less than 1%

in 4.1D, and 0% in 4.1E. The spectrum labeled 6 has a 0% probability of occurring

in 4.1A, 0.1% in 4.1B, 1% in 4.1C, 8% in 4.1D and > 55% probability in 4.1E.

The evolution of these gapmaps (Figure 4.1) with falling doping, from domination

by heterogeneous but predominantly superconducting spectral characteristics (Figure

4.1A,B) to domination by spectra of a very different type (Figure 4.1E) is striking.

Despite the intense changes with doping in the gapmaps (whose information

content is, by definition, derived from the coherence peaks at E = ∆(~r)), the LDOS

at energies below about 0.5∆ remains relatively homogenous for all dopings studied.

Figure 4.1F reveals this low-energy LDOS homogeneity because, independent of gap

value, the g(E) below ≈ 25 meV are almost the same everywhere and for all spec-

tra. These low energy LDOS do, however, exhibit numerous weak, incommensurate,

energy-dispersive, spatial LDOS-modulations with long correlation lengths (for ex-

ample Figure ??C). We focus on the doping dependence of these low energy g(~r, E)

data by applying the FT-STS technique. Figure 4.2A-C shows measured g(~q, E)

for the three g(~r, E) datasets used to generate Figure 4.1A, 4.1D, and 4.1E. Each

sub-panel is the measured g(~q, E) at the labeled energy, with the reciprocal space

locations of the Bi (or Cu) atoms ~q = (±2π/a0, 0) and (0,±2π/a0), appearing as the

four dark spots at the corners of a square. It is obvious that multiple sets of dispersive

LDOS-modulations exist at all three dopings, but each exhibits different trajectories

as a function of E for different p.

Analysis of these low energy LDOS modulations requires a model for their re-

lationship to states in ~k-space. We apply the ”octet model” of quasiparticle interference[70,

50] which is predicated on a Bi-2212 superconducting band-structure exhibiting four

sets of ’banana’-shaped contours of constant quasiparticle-energy surrounding the gap

nodes[25]. Because of the quasiparticle density of states at E is

n(E) ∝∮

E(k)=ω

1

∇kE(~k)dk (4.1)

45

89KOD 74KUD 65KUDa) b) c)

Figure 4.2: (a)-(c) Examples of measured g(~q, E) for a variety of energies E as shown at

three doping levels.

46

89KOD 74KUD 65KUD

5 10 15 20 25 30 35 40

0.9

1.0

E (meV)

q1

q5 89KOD

q7

q1

q5 74KUD

q7

q1

q5 65KUD

q7

|q|

(2π/a

0)

c)b)

ky (

2π/a

0)

kx (2π/a0)

0 0.20.1 0.3 0.4 0.50

0.8

0.7

0.6

0.5

0.4

0.0

0.1

0.2

0.3

0

0.2

0.1

0.3

0.4

0.5

Figure 4.3: (a) A schematic representation of the 1st Brillouin zone and Fermi surface

location of Bi-2212. The flat-band regions near the zone face are shaded in blue. The eight

locations which determine the scattering within the octet model are show as red circles

and the scattering vectors which connect these locations are show as arrows labeled by the

designation of each scattering vector. (b). Measured dispersions of the LDOS-modulations

~q1, ~q5 and ~q7 for the 3 dopings whose unprocessed data is shown in Figure 4.2. We chooses

this set of three ~q-vectors because they exhibit the maximum intensity of any set sufficient

to independently determine the locus of scattering ~ks(E) for all dopings. (c). Calculated

loci of scattering ~ks(E) for all 3 dopings. The blue line is a fit to the 89KOD data.

while each ‘banana’ exhibits its largest |∇kE(~k)|−1 near its two ends, the primary

contributions to come from the octet of momentum-space regions at the ends of each

‘banana’ ~kj; j=1,2,..8

(Figure 4.3A). Mixing of quasiparticle states in the octet by disorder scatter-

ing produces quasiparticle interference patterns which are manifest as spatial LDOS-

modulations. The intensity of such scattering induced modulations is primarily gov-

erned by joint density of states (among other factors). The wavevectors of the most

intense LDOS-modulations are then determined by all possible pairs of points in the

octet ~kj. Sixteen distinct +~q and −~q pairs should be detectable at each non-zero

47

energy by FT-STS. From them, the energy dependence of the octet locations ~kj(E)

can be determined and associated with a ’locus of scattering’ ~ks(E). Comprehen-

sive internally-consistent agreement between Bi-2212 STM data and this model is

achieved near optimal doping[50]. Until this work, nothing was known about its

utility for strongly underdoped cuprates.

Theoretical analyses beyond the simple octet model[71, 40, 53, 39, 72, 73, 74,

75, 76, 77] capture many elements of our previously reported g(~q, E) data, but no

resolution of the exact source, strength, or type of scattering has yet been achieved.

Nevertheless, the existence of numerous sets of long-correlation length, dispersive,

LDOS modulations, all of which are self-consistent with a single ∆(~k) for both filled

and empty states, is indicative of good Bogoliubov-like quasi-particles. Since the

LDOS-modulations can be associated consistently with a ’locus of scattering’ ~ks(E)

via the octet model, we analyze our observations within this model using the ~q-vector

designations shown in Figure 4.3A.

Figure 4.3B shows the measured length of ~q1, ~q5 and ~q7 as a function of en-

ergy for the three datasets in Figure 4.2. Figure 4.3C shows the locus of scattering

calculated for these three using:

~q1 = (2kx, 0); ~q5 = (0, 2ky); ~q7 = (kx − ky, ky − kx) (4.2)

~ks = (±kx(E),±ky(E));~ks = (±ky(E),±kx(E)) (4.3)

The ~ks(E) determined by this technique differs only slightly between dopings. even

though the actual g(~r, E) for different dopings are quite different at any given energy.

These three ~ks(E) are each the same for filled and empty sates and each consistent

with the same ∆(~k) at that doping. Thus Bogoliubov-like quasi-particles appear to

exist at these momentum space locations at all dopings. This is consistent with the

small motion of the FS in this doping range detected by ARPES.

These observations certainly do not exhaust the changes observed in g(~r, E)

with falling doping. A very strong effect is the evolution, with doping, of the ~q-space

location of strongest LDOS-modulation at any energy. This modulation is always

48

associated with ~q1 and its location evolves from ~q1 = 2π6a0

at p= 0.19±0.01, to ~q1 = 2π5.1a0

at p= 0.14 ± 0.01, to ~q1 = 2π4.7a0

at p= 0.10 ± 0.01. Another effect is a decrease in

relative intensity of dispersive LDOS-modulations ~q2, ~q3, ~q6, ~q7 relative to those of

~q1, ~q5, with falling with p.

The doping dependence of states with ~k = (±π/a0, 0) and (0,±π/a0) in the

’flat band’ region near the zone-face[25] (green shaded areas in Figure 4.3A) is ex-

tremely different. These states can also be identified by FT-STS analysis of g(~r, E)

data. By definition, the coherence peaks in g(~r, E) occur at E = ∆(~r). In all samples,

they exhibit intense particle-hole symmetric LDOS-modulations, with wavevectors

~G = (±2π/a0, 0) and (0,±2π/a0)[50]. These coherence peak LDOS-modulations

at E = ∆ possibly occur due to Umpklapp scattering between ~k = (±π/a0, 0) and

(0,±π/a0)[50]. Therefore, the coherence peaks in tunneling are identified empirically

with the zone-face states at ~k = (±π/a0, 0) and (0,±π/a0). This identification is

also consistent with theory. The coherence peaked tunneling spectra (e.g. Fig ??F:

spectra 1-4) are theoretically viewed as due to superconducting pairing on the whole

FS because such spectra are consistent with a ∆x2−y2 everywhere on the ARPES-

determined FS near optimal doping[20]. We therefore consider any spatial regions

that show clear coherence peaks with ~q = ~G LDOS-modulations, to be occupied by

a canonical d-wave superconductor (dSC).

Near optimal doping, more than 98% of any FOV exhibits this type of co-

herence peaked dSC spectrum. As the range of local values of ∆(~r) increases with

decreasing doping, the intensity of the ~q = ~G coherence peak LDOS-modulations

becomes steadily weaker until, wherever ∆(~r) > 65 meV, they disappear altogether.

This process can be seen clearly in the gap-averaged spectra of Figure 4.1F where the

average height of the coherence peaks declines steadily with increasing ∆. It is found

equally true for all dopings. Wherever the coherence peaks and their ~q = ~G LDOS-

modulations are absent, a well-defined new type of spectrum is always observed.

Figure 4.4A shows a high resolution gapmap from a strongly underdoped sample.

49

-100 50 1000.00

0.05

0.10

0.15

0.20

0.25

0.30

0.35

0.40

0.45

0.50

q1q*

dI/d

V (

nS

)

b)

c)

Sample Bias (mV)

|q|

(2π/a

0)

-400 -200 0 200 400

0.5

∆ > 65 meV

∆ < 65 meV

d)

Sample Bias (mV)

q*

-50 0

a)

0.4

0.3

0.2

0.1

0.0

Figure 4.4: (a) A high-resolution 46nm square gapmap from a strongly underdoped sample.

(b) Examples of representative spectra from (1) nanoscale regions exhibiting coherence

peaked spectra with ~q = ~Q LDOS-modulations at E = ∆(~r) (red) and (2) from regions

exhibiting ZTPG spectra (black). The locations where these spectra occur are shows as

small dots in (a). (c) The mask identifying regions with ∆ <65 meV from the ZTPG

regions calculated from gapmap in (a). (d) Dispersion of ~q1(E) in regions with coherence

peaked spectra ∆ < 65 meV is shown in red. There are no modulations at any higher

energies within our range. Dispersion of ~q1(E) in regions with ZTPG spectra for E< 36

meV (black symbols). The red squares in Figure 4.3(b) represent a combination of these

two (indistinguishable) dispersions in this range. For E> 65 meV, the wavevector of the

new modulations in ZTPG regions are shown in black. To within our uncertainty they do

not disperse and exhibit ~q∗ = (±2π/4.5a0, 0) and (0,±2π/4.5a0)± 15%.

50

Examples of this new type of spectrum, along with those of coherence peaked dSC

spectra, are shown in Figure 4.4B. The coherence peaked spectra (red) are manifestly

distinct from the novel spectra (black) which have a V-shaped gap reaching up to -300

meV and +75 meV. For reasons to be discussed below, we refer to the new spectrum

as the zero temperature pseudogap (ZTPG) spectrum.

The replacement of coherence peaked dSC spectra by ZTPG spectra first be-

gins to have strong impact on averaged properties of g(~r, E) and g(~q, E) below about

p=0.14 where the fractional area covered by ZTPG spectra first exceeds 10% of the

FOV. In terms of the spectral shape no further evolution in the form of the ZTPG

spectrum is detected at lower dopings. Instead, a steadily increasing fractional cov-

erage of the surface by these ZTPG spectra is observed. Our previous studies[48, 49]

were carried out at dopings p > 0.14 and, when ZTPG spectra have been detected

at such higher dopings,[48, 45] it is in a tiny fraction of the FOV. Very significantly,

spectra similar to the ZTPG spectrum are detected inside cores of Bi-2212 quantized

vortices where superconductivity is destroyed[78, 79]. Furthermore, a very similar

spectrum is observed in another very underdoped cuprate NaxCa2−xCuO2Cl2[64, 29],

even in the non-superconducting phase. It therefore seems reasonable that the char-

acteristic zero-temperature spectrum in the pseudogap phase is of this type. This is

why we tentatively assign this spectrum the ZTPG designation.

4.2 FTSTS and Doping

As discussed in chapter 2, the g(~r, E) for E < ∆/2 in even the most underdoped

samples (Fig 4.1E, Figure 4.2C) exhibit relatively homogenous electronic structure

with good quasi-particles dispersing on the Fermi-arc. However, for E > ∆/2 in these

same samples, our previous analyses techniques fail, probably because very different

phenomena are occurring in different nanoscale regions of each FOV. To explore the

implications of the ZTPG spectrum for strongly underdoped samples, new analysis

techniques are therefore required. Here we introduce a masking process which has

51

proven highly effective. From a given strongly underdoped data set, the g(~r, E) in

all regions where E∆ > 65 meV are excised and used to form a new masked data set

g(~r, E)|∆>65. The remainder forms a second new dataset g(~r, E)|∆<65. The E∆ > 65

meV cutoff was chosen because, on the average, it represents where the coherence

peaks with associated ~q = ~G modulations have all disappeared and are replaced by

the ZTPG spectra. An example of this type of mask for the gapmap in Figure 4.4A

is shown in Figure 4.3C. It is important to note a serious drawback of the masking

process. The ~q resolution is of masked data is considerably worse in than those shown

in Figure 4.3B because the largest contiguous nano-region in the mask is about 20%

of the full FOV. As a result, the precise modulation period and dispersions of any

effects detected by masking cannot can be determined with nearly the same degree

of accuracy as the low-energy quasi-particle interference signal (Figure 4.3 and Ref.

28).

4.2.1 Charge Order and Underdoped Bi2Sr2CaCu208+δ

FT-STS analysis of such (g(~r, E)|∆<65, g(~r, E)|∆>65) pairs shows that they exhibit

dramatically different phenomena. In the g(~r, E)|∆<65, the dispersive trajectory of ~q1

is seen up to E ≈ 36 meV and no further LDOS-modulations can be detected at any

higher energy (red symbols in Figure 4.4D). In the g(~r, E)|∆>65 data, the identical

dispersive ~q1 signal is seen below E≈ 36 meV. However, a new LDOS-modulation

appears in the g(~r, E)|∆>65 between E> 65 meV and our maximum energy E=150

meV (black symbols in Figure 4.4D). We designate its wavevector ~q∗.

To explore the real-space structure of this new high-energy LDOS-modulation,

we define a map Γ15065 (~r) = Σ150

E=65g(~r, E)|δ>65 which sums over this energy range. This

map is shown in Figure 4.5A and, although it has quite a disordered mask, careful

examination reveals checkerboard modulations within each nano region. Importantly,

Fourier transform analysis of this Γ15065 (~r) shown in Figure 7B reveals a well-defined

wavevector set ~q∗ = (±2π/4.5a0, 0) and (0,±2π/4.5a0) ± 15% for these new high-

52

a)

b)

0 0.25 0.5 0.75 1.0

FT

In

ten

sity

(a

rb)

|q| (2π/a)

c)

FT

In

ten

sity

(a

rb)

0.25 0.5 0.75 1.0

∆<65 meV

∆>65 meV

|q| (2π/a0)

∆<65 meV

∆>65 meV

FT

In

ten

sity

(a

rb)

0.20 0.21 0.22 0.23 0.24 0.25 0.26 0.27

|q| (2π/a0)

d)

Figure 4.5: The image of g(~r, E) masked by Figure 4.4c) and then summed from 65 meV

to 150 meV :Γ15065 (~r) = Σ150

65 g(~r, E)|δ>65. Regions within the mask show ‘checkerboard’

modulation. The complimentary map is featureless except crystalline effects. b) the Fourier

transform of a). The Bi atom locations are circled in orange. The inset shows the plot

of the Fourier transform amplitude along the line shown. It reveals a maximum at ~q∗ =

(±2π/4.5a0, 0) and (0,±2π/4.5a0)±15% as indicated by the blue arrow. (c). The magnitude

of the Fourier transform of the masked topographic image, taken at 150 mV and 150 pA,

along the ~qtopo||(2π, 0) direction is modulated with ~qtopo = (±2π/5a0, 0) and (0,±2π/5a0)±25% (black squares). Fourier transforms of the complementary part of the topography

(from < 65 meV regions not exhibiting ZBTG spectra) shows no such modulations at any

wavelengths near this ~qtopo (red triangles). The lines are guides to the eye. The difference

between these two Fourier transform intensities is shown in blue and shows topographic

modulations in the ZTPG regions that are undetectable elsewhere. d). A plot of the

amplitude of the ~q1 LDOS modulation as a function of |~q1| for the sample data as in Figure

4.1c), 4.2e), 4.3(d). The maximum intensity of the modulations in the ZTPG regions occurs

at |~q1| = 2π/4.8a010%. No special scattering is observed of the quasiparticles in the dSC

regions.53

energy modulations as indicated by the arrow on black data points in inset to Figure

7B. The identical analysis the complementary map Σ15065 g(~r, E)|δ<65 is featureless near

~q∗ (red data in inset of Figure 7B). Thus, an LDOS-modulation with very low (or

zero) dispersion, exists above ±65 meV exclusively in regions characterized by ZTPG

spectra of strongly underdoped Bi-2212 samples.

Constant-current topography represents, albeit logarithmically (see chapter

??), the contour of constant integrated density of states up to the sample-bias en-

ergy. It does not suffer from the systematic problems due to effects of the constant-

current setup condition renormalization[48] which plague g(~r, E). It is therefore

a more conclusive technique for detection of net charge density modulations by

STM. To search for topographic modulations, we apply the identical mask (Fig-

ure 4.5C) to the topographic image which was acquired simultaneously with the

gapmap in 4.4A. The magnitude of the Fourier transform along the ~q||(2π, 0) for the

masked topographic image shows that, in the ZTPG regions, the topography is mod-

ulated with ~qtopo = (±2π/5a0, 0) and (0,±2π/5a0)± 25%(indicated by the arrows in

Fig 4.5D). Fourier transforms of the complementary part of the topography (from

∆ <65 meV regions not exhibiting ZBTG spectra) shows no such modulations at

any wavelengths near this ~qtopo (red in Figure 4.5C). No topographic modulations

near ~qtopo are detected anywhere in samples at higher doping. An additional peak

near ~qrecon = (2π/2a0,±2π/2a0) and (−2π/2a0,−2π/2a0) in the Fourier transform of

the topograph comes from a reconstruction along the supermodulation maximum, is

observed at all dopings and, since no signature is observed in LDOS at ~qrecon for any

energy or doping, is regarded as irrelevant.

Some important new facts about the strongly underdoped regime of Bi-2212

electronic structure emerge from these results. Our first finding is related to the

Fermi-arc quasi-particle states. As shown in Figure 5C, FT-STS indicates that quasi-

particle interference occurs between Bogoliubov-like states in approximately the same

region of ~k-space for all dopings. These Fermi-arc quasiparticles remain spatially ho-

54

mogenous (except for relatively weak LDOS-modulations) even in the most under-

doped samples. They are Bogoliubov-like in the sense that they exhibit particle-hole

symmetry at each location in ~k-space, and, at each doping, are all consistent with

the same ∆(~k). Therefore, one can reasonably postulate that Fermi-arc states are

gapped by superconducting interactions at all dopings studied. If so, they, and the

associated ’nodal’ superconducting state[80, 81, 82, 83], are amazingly robust against

the heterogeneous electronic phenomena which so dominate Bi-2212 at other energies.

Our second finding is the very different fate of states in the flat-band re-

gions near ~k ≈ (±π/a0, 0); (0,±π/a0). The appearance of ZTPG spectra in strongly

underdoped samples coincides exactly with destruction of antinodal superconduct-

ing coherence peaks. Exclusively in these ZTPG regions (black in the gapmap),

we find the three new modulation phenomena: (1) topographic modulations with

~qtopo = (±2π/5a0, 0) and (0,±2π/5a0) ± 25%, (2) a peak in LDOS centered around

~q∗ = (±2π/4.5a0, 0) and (0,±2π/4.5a0) ± 15% for E >65 meV to at least E=150

meV and, (3) the dispersive ~q1 quasi-particle branch exhibits its maximum modula-

tion intensity when it passes through ~q1 = ~qtopo = ~q∗ (Fig 7D). This last situation

has been predicted[72, 75, 39, 73, 84] as a consequence of the Fermi surface geometry

and quasi-particle dispersion in the presence of potential scattering from charge-order

with fixed ~Q; in this case ~Q = ~qtopo = ~q∗. Taken together, these observations all point

to the appearance of an unusual charge ordered state with ~q∗ = (±2π/4.5a0, 0) and

(0,±2π/4.5a0)±15%, occurring only in the regions characterized by the ZTPG spec-

trum and only in strongly underdoped Bi-2212. Note that due to the strong disorder

and the limited size of the mask domain we can not distinguish between a charge

density modulation caused by a condensed charge order and that caused by impu-

rity through an enhanced charge susceptibility in a state without condensed charge

order[85]. In addition we do not imply that there is true charge long range order

since from the Imry-Ma argument this is always absent in a disordered system like

Bi-2212.

55

Charge order has been observed in other underdoped cuprates, including Nd

doped La-214 with inelastic neutron scattering[86] and more recently by STM in

NaxCa2−xCuO2Cl2[29]. It has also been proposed, based on reported nondispersive

(between 0 and 20 meV), ’line object’, LDOS modulations with ~q = (2π/4a0, 0)[51],

that static stripes exist in optimally doped Bi-2212 below Tc. However, none of these

phenomena have been detected in several independent higher resolution studies[70,

50, 87].

Other very suggestive findings have also been made by STM in Bi-2212. Field

induced sub-gap LDOS-modulations, with ~qvortex = (±2π/4.3a0, 0) and (0,±2π/4.3a0)±15% were discovered surrounding vortex cores (where superconductivity is destroyed)

near optimal doping. This observation provided the first STM evidence for some

type of incipient charge-order competing with superconductivity in cuprates[88]. Pi-

oneering STM experiments to map the low energy LDOS above Tc when supercon-

ductivity is also destroyed, have detected sub-gap LDOS-modulations with ~qPG =

(±2π/4.6a0, 0) and (0,±2π/4.6a0)[87] at near-optimal doping. Although neither of

these low energy phenomena are fully understood, they do, along with low energy

phenomena in ZTPG regions reported here, form a triad of apparently consistent

observations. Destruction of superconductivity, whether by high magnetic fields, by

exceeding Tc, or by strong underdoping, results in very similar effects on low energy

LDOS modulations. It remains to be determined how the vortex core and pseudogap

observations relate directly to the charge order.

4.3 Implications and Conclusions

The identity of the electronic phase represented by the ~qtopo = ~q∗ charge order

is difficult to discern. In the absence of disorder, plaquette orbital-order phases

such as staggered flux phase (SFP)[89, 90] and D-density-wave (DDW)[91], or intra-

plaquette orbital phases[92], are not expected to exhibit topographic or LDOS-

modulations. However, in theory, LDOS-modulations can be produced by vortex

56

and disorder scattering in the SFP and DDW phases but not near ~q = (±2π/4a0, 0)

and (0,±2π/4a0)[76, 77, 93]. It remains theoretically unexplored whether disordered

orbital phases could result in the complete set of new phenomena (see below) we report

here. Charge-ordered phases including stripes[94, 95, 96], disorder pinned electronic

liquid crystal[52], strong-coupling spin- and charge-density waves[97], and, recently,

hole-pair crystals[98] have been proposed to exist in underdoped cuprates. Each of

these phenomena would yield both LDOS- and topographic-modulations. But again,

it remains theoretically unexplored whether these theories can account for our obser-

vations that in the ZTPG regions (i) a characteristic new tunneling spectrum exists,

(ii) the ~q∗ modulations appear only above a relative high energy (≈ 65 meV), (iii) they

exhibit an incommensurate wavevector ~q = (±2π/4.5a0, 0) and (0,±2π/4.5a0)±15%,

(iv) they exhibit the same spatial phase for positive and negative biases - so that the

filled-state density maxima coincide with the empty-state-density maxima, and (iv)

the ~q∗ modulations are replaced by the dispersive quasi-particle interference signals

at sub-gap energies. A further point is that the predicted strong breaking of the 90

rotation symmetry in the stripe scenario is not observed in any of the STM studies,

but again, it may be possible that this is due to the presence of strong disorder. For

all these reasons the precise identity of the charge-order state in strongly underdoping

Bi-2212 remains elusive.

The data reported here also motivate a new conjecture on the evolution of elec-

tronic structure with reduced doping in Bi-2212. In ~r-space, we identify two extreme

types of LDOS spectra (Figure 6B). The first exhibits clear coherence peaks at the

gap edge and dominates near-optimal samples. The second type (ZTPG-spectrum)

exhibits a V-shaped gap over a much wider energy range and dominates in strongly

underdoped samples. We associate the former with a pure d-wave superconducting

state and conjecture that the latter reflects a zero temperature charge ordered state

existing at sufficiently low dopings in the pseudogap regime. If the whole Bi-2212

sample were homogeneous and consisted of only one of the above phases, then, in

57

~k-space, quasi-particle peaks would exist all along the Fermi surface in the pure d-SC

phase, but only on a finite arc around the gap nodes in the charge ordered phase with

the zone face states being incoherent due to localization. This may indeed be the case

in NaxCa2−xCuO2Cl2[29, 60]. In Bi-2212 he reality is more complicated. In optimally

doped samples, more than 98% of the surface area exhibit dSC spectra (Figure 3A,B).

As doping falls the ZTPG regions appear and grow in significance until as doping ap-

proaches p ≈ 0.1, almost 60% of the area exhibits the ZTPG characteristics and the

associated charge-order (Figure 3E). From this trend, it is reasonable to expect that,

at even lower doping in the zero temperature pseudogap phase, 100% of the sample

would exhibit the ZTPG characteristics and be charge-ordered.

In a spatially disordered situation, the probability of occurrence of the two

types of phenomena evolves continuously with doping in a fashion related to the evo-

lution of the gapmaps in Figure 3. Therefore, properties which average over nanoscale

phenomena would appear to evolve smoothly between the two extremes. Experi-

mental results which are relevant to this proposal include, for example, the doping

dependence of the ARPES (π, 0) peak[21, 25, 33, 57], the specific heat jump at the

superconducting phase transition[21], and the c-axis conductivity[21]. Due to the

heterogeneous mixture of the dSC and the ZTPG regions, it is difficult for a spatially

averaged experiment like ARPES to discern the properties of the ZTPG region. How-

ever, the well-known tendency of the coherent quasiparticle peaks near the zone face

to be suppressed by underdoping[25, 33, 57], is consistent with our conjecture. The

specific heat jump at the superconducting transition is a characteristic specifically

of a dSC phase and not of the charge-ordered ZTPG phase. Hence, the declining

specific heat jump as a function of underdoping[21, 99] is also consistent with our

hypothesis. Finally, due to tunneling matrix element effects, c-axis tunneling senses

the zone-face quasiparticles instead of the nodal ones. Hence the decrease of the

c-axis conductivity with underdoping[21] also seems consistent with our suggestion.

Further inter-comparison between the results of these experimental techniques will

58

be required to explore these proposals.

59

Chapter 5

Sources of Disorder in

Bi2Sr2CaCu208+δ

The stoichiometric parent compounds to the cuprates are insulators. Unlike band

insulators, the driving force for this localization is Coulomb repulsion and stoichiom-

etry not the Pauli exclusion principle[18, 20]. In order for the electrons to become

itinerant some of them need to be removed (or added). To do this “dopant” atoms

are added to move the electron occupation to a fractional number.

This means that in all high-Tc compounds, when they are superconducting

nonperiodic potentials must be present. Thus disorder is inherent to these systems.

This dopant disorder seems like an obvious candidate as the cause of the gapmap

disorder[42, 43, 44, 45, 46, 47, 48, 49].

5.1 Gapmap and the Question Why

It has long been proposed that the dopant atoms in Bi2Sr2CaCu208+δ is the cause

of the gapmap disorder[100, 48, 101, 102]. But the exact mechanism by which the

dopants effect the low energy electronic structure is still debated.

One such proposal relies on the dopant atoms themselves being charged and

thus responsible for a Coulomb potential which can attract the doped holes in the

60

CuO planes. The random distribution of these attracting potentials could result in

a local variation in doping[100, 48, 101]. Since the average properties are known to

depend on average hole concentration in the CuO planes this doping disorder could

directly lead to variations in local observables like ∆(~r) .

Another proposal treats the dopant atoms as scattering sites of a particular

type. Since these additional atoms reside in other planes than the CuO they would

result small angle scattering[103]. Numerical simulations of the effect on this weak

scattering has resulted in spatial variations of the gap magnitude[102].

A final proposal involves the small scale strains that these interstitial and

substituted atoms would add the the crystal. It was found that the local strain was

strongly coupled to local order parameter variation[104].

The first step in deciding between theses different proposals is to find the

dopant atoms. But since they are not strongly coupled to the CuO states, they

cannot be found directly by using low energy spectroscopic methods.

5.2 Photoemission

For over a decade Photoemission experiments have studied the cuprates and can tell us

much about their electronic structure. Since the early 1990s numerous groups have re-

ported cuprate states that reside about 1 eV below the Fermi energy[105, 106]. These

were thought to be associated with the delocalized Zhang-Rice (ZR) triplets[107]. By

measuring the ~k dependence of these states Sawatsky and collaborators[108] found

that at least part of these “1 eV peaks” were not of the same symmetry as these ZR

states. Instead they found that they were most likely due to O 2p states which have

little or no hybridization with the Cu 3d states.

61

5.3 Discovery

To investigate if these nonhybridized states were really associated with localized states

STM conductance spectra were taken out to 1 eV. As and aside, at 1 V bias and 1

GOhm junction resistance the electric field between the tip and sample is upwards of

a factor of 10 and even 100 greater than what is usually used for STM data taking.

This increased electric field places much greater constraints on tip preparation and

a great number of tips that are excellent for low Voltage studies fail under these

conditions.

5.3.1 Spectral Characteristics

Conductance spectra were taken between between -1.1 and 1.1 mV. A large broad

feature consisting of a peak centered near -950 mV about that is 150 mV wide is seen

in some spectra. Examples spectra with and without the 1 eV peak are shown in

figure 5.1 (red and black respectively). A large broad peak is easily seen in the red

spectrum.

To investigate the density of these objects g(~r, V ) was measured over 49 nm

FOVs on multiple samples of various dopings. Two of these datasets are shown in

figure 5.2. Above each is the concurrent topograph.

The number of objects in each of the g(~r, V ) maps is noticeably different.

Comparing them to the nominal doping is difficult in Bi2Sr2CaCu208+δ . This is be-

cause BiSrCaCuO-2212 is not really the chemical composition. Instead 10% of the Sr

sites are filled with Bi making the chemical makeup closer to Bi2.2Sr1.8CaCu208+δ. To

avoid this ambiguity, we rely on average gap of the samples (as mentioned in the pre-

vious chapter). ARPES[25] break-junction tunneling[69], and thermal conductivity[?]

have all found that the average gap, ∆, steadily decreases with increased doping.

By plotting ∆ versus twice the number density of these objects (2 holes for

every O 2p shell filled), we find that they behave approximately as expected. This can

62

a) b)

Figure 5.1: a) Two spectra on and off the “1 eV feature” (red and black respectively). b)

The shape of the feature for 4 different regions, found by imaging the conductance at -950

mV.

be seen as the black squares in figure 5.3. The black line is taken from Sutherland et

al 2003, and approximates the average gap as seen by thermal conductivity in YBCO

and by ARPES in Bi2Sr2CaCu208+δ . Clearly the number density of these ‘-960

meV’ peaks scales correctly with doping except for a systematic 5% deficit (the red

squares show the data offset by 5%). The cause of this 5% is unknown. Because these

atomic-scale ’-0.96V features’ are found at (i) the expected energy for un-hybridized

oxygen 2p orbitals, (ii) in the correct crystal locations of the BiO layer, and (iii) with

densities varying approximately as expected with bulk doping, we propose they are

the oxygen dopant-atoms.

5.3.2 Location of the Oxygen Atoms

Although they do have a typical position in the crystal, in the center of the BiO

plaquette, there is no ordering to the Oxygen atoms.

To test if they are randomly distributed, histograms of their distances to each

63

a) b)

c) d)

50nm

Figure 5.2: a) Two different g(~r, V ) maps at V = -900 mV with their associated topographs

above. The setup conditions for these data were 30pA and -900 mV and a bias modulation

of 50 mV RMS was used.

64

0.00 0.05 0.10 0.15 0.20 0.25

0

20

40

60

80

100

Doping by counting O atoms

Doping shifted by +0.05

Gap Vs. Doping from Sutherland

et al and ARPES (approx)

∆ (

me

V)

p (# per Cu)

Figure 5.3: a) The average gap versus twice the number density of oxygen atoms for 3

samples (black squares). The red squares are simple the data shifted horizontally by %5.

The black line is taken from Sutherland et al [?] and represent ∆ as a function of doping as

measured by thermal conductivity and ARPES.

65

0 200 400 600 800 1000 1200 1400

0.0

0.2

0.4

0.6

0.8

1.0

pro

babili

ty o

f a g

iven r

adiu

s (

norm

aliz

ed)

radius (pixels)

~ 45 meV

~ 55 meV

Random

Figure 5.4: a) The Histogram of distances between oxygen atoms for the two samples in

figure 5.2 and a random distribution of points.

other were made. These were then compared to the histogram from a random distri-

bution of points. These histograms are shown in figure 5.4, showing no differences at

any length scale. Thus they are more or less randomly distributed around the BiO

plane.

5.4 Correlation with Other Observables

After identifying them the next task is to find their effect on the low energy properties

of Bi2Sr2CaCu208+δ . To do this low energy spectroscopic maps were taken in the

66

same FOVs as oxygen maps. The correlations were investigated.

5.4.1 Definition of Terms

To compare the correlation between multiple observables one way is to look at the

correlation coefficient, Cf,g. To get to the definition of this one let’s do a little defining

of terms. For two functions f and g. I’ll define the zeroed autocorrelation, Af,f (R),

of a function, f , as

Af,f (R) =

∫ ∞

−∞[f(r)− f ]× [f(r + R)− f ]dr, (5.1)

where f is just the average value of f (this makes the autocorrelation go to zero at

infinity). The cross correlation A of two functions f and g is then

Af,g(R) =

∫ ∞

−∞[f(r)− f ]× [g(r + R)− g]dr, (5.2)

This cross correlation has the units of [f ]× [g] so it doesn’t show the degree to which

two functions are correlated independent of this. To fix this you divide this cross

correlation by the geometric mean of the zero value of the autocorrelation [Af,f (R =

0)× Ag,g(R = 0)]1/2. This then gives the cross correlation coefficient Bf, g as

Cf, g(R) =Af,g(R)

[Af,f (R = 0)× Ag,g(R = 0)]1/2. (5.3)

This is formally a functions of R. I am still not sure if the functional form of this has

any real meaning so I will really focus on the R = 0 value of this coefficient. It’s value

is limited to (−1, 1) for perfectly anti-correlated to perfectly correlated. For now we

only look at the absolute values to see which observables effect which others.

5.4.2 Results Sample ∆ = 45 meV

The observables we are interested in comparing are 4 fold. These are a) the map of

the oxygen atoms: O(r), b) the high voltage topography taken at −900 meV Th(r),

c) the high voltage topography taken at −100 meV Tl(r), d) the gapmap ∆(r). An

67

a) b)

c) d)

Figure 5.5: a) The oxygen map, O(r): conductance map at −900 mV with a ±100 meV

window. b) The high energy topography, Th: setup parameters of 30 pA and −900 mV. c)

The gapmap, ∆(r): color-scale of red is 20 meV and black is 70 meV and greater. d) The

high energy topography, Tl: setup parameters of 100 pA and −100 mV. All of these are in

the same field of view in the sample from run c10.

68

0 20 40 -1.0 -0.9 -0.8 -0.7 -0.6 -0.5 -0.4 -0.3 -0.2 -0.1 0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0

Cor

rela

tion

Coe

ffici

ent (

betw

een

0 an

d 1)

Distance (Å)

Oxygen cross Topo(-900mV) Oxygen cross Topo(-100mV) Oxygen cross Gapmap Topo(-900mV) cross Topo(-100mV) Topo(-900mV) cross Gapmap Topo(-100mV) cross Gapmap

Figure 5.6: (a) Angular average of the normalized cross correlation for the combinations

of observables shown in figure 5.5 with ∆ = 45 meV.

example of these is shown in figure 5.5. There are n(n−1)/2 correlation functions for

n observables. For the four observables in figure 5.5 represented by the 6 black line

segments. Calculating the correlations coefficients and taking their angular average

(r is now a scalar although above it was not) results in figure 5.6.

The correlations of the two topographs shows that the data is all lined up well

but is otherwise trivial. Clearly ∆(r) and Tl(r) are the best correlated in this field of

view. But second is the oxygen map, O(r), with Tl(r) and ∆(r) which are correlated

with about the same magnitude. The high energy topography does not correlate well

with either ∆ or Tl, because by −900 mV the inhomogeneity is no longer present in

69

0 20 40 -1.0

-0.8

-0.6

-0.4

-0.2

0.0

0.2

0.4

0.6

0.8

1.0

Cor

rela

tion

Coe

ffici

ent (

betw

een

0 an

d 1)

Distance (Å)

Oxygen cross Topo(-900mV) Oxygen cross Topo(-100mV) Oxygen cross Gapmap Topo(-900mV) cross Topo(-100mV) Topo(-900mV) cross Gapmap Topo(-100mV) cross Gapmap

Figure 5.7: Angular average of the normalized cross correlation for the combinations of

observables shown in figure 5.5 but for the sample with ∆ = 55.

the topography.

5.5 Results ∆ = 55

The correlation coefficients for the underdoped sample from run c8 are shown in figure

5.7. The correlations of the two topographs shows that the data is all lined up well but

is otherwise trivial. Things are very similar in this sample to the Ni sample. Except

for two things. This first and least interesting is that the low energy topograph is less

correlated with the gapmap. This might be due to the reduced quality of this data.

70

The other is how poorly the oxygen map correlates with the low energy topograph.

The previous rendering with this one being really high was in error.

5.6 Conclusion

Because oxygen mapping is still a work in progress many of the conclusions are

still speculative. But nonetheless several important new insights into atomic-scale

HTSC electronic structure emerge from these studies. First, the dopant oxygen 2p

orbital can be imaged at -0.96V for the first time and correlated with local electronic-

structure. Second, the source of the scatting leading to QI source and the anomalous

intensity of q1 are strongly due to the oxygen dopant atoms. Third, strong correla-

tions between low-bias topo and gapmap are due to spectral weight shifting and not

hole(electron)-accumulation with charge density variations. Perhaps equally impor-

tant is the demonstration of a technique to directly determine the impact of individual

dopant atoms on electronic structure. It can help in engineering the doping process

so as to avoid the disorder and increase electrical/thermal conductivity and possibly

even the HTSC critical temperatures. Further, these techniques are likely to prove

equally useful for equivalent studies on other exotic doped insulators.

71

Appendix A

Atomic resolution and STM

evaluation

A.1 On the Topic of Homogeneity

For well over a decade research groups that measure the tunneling spectra on

Bi2Sr2CaCu208+δ have reported large variations of the tunneling spectra of states.[42,

43, 44, 45, 46, 47, 48, 49] One other group claims to be able to achieve homogeneous

samples through oxygen annealing.[109] Before resolving this debate standards for

STM data are needed.

Since an STM scans a surface with atomic resolution the natural assumption is

that the measurements see sub-nanometer features. As long as the atomic periodicity

is seen in topographic data, “atomic resolution” is claimed and the data is taken as

accurate. Simple models of STM tips, where the tip is pointed or has a hemispherical

tip of a set radius, help to enforce these perceptions.[28] In these models the radius

of curvature of the STM tip limits the radius of curvature of any features that can be

resolved on the surface.

In reality STM tips can have very complicated structures. Since the measured

topography is the convolution of the tip and surface shape, when the tip has any

72

a) b)

(0,2π/a0)

(2π/a0,0)

Figure A.1: a) A topography on a slightly overdoped surface. The atomic periodicity

can be weakly resolved. b) The Fourier transform of a topography with the sam tip and

surface. The atomic periodicity can clearly be seen in the corners of the image and are

labeled (2π/a0,0) and (0,2π/a0).

feature with a small radius of curvature then the atomic lattice will still show through.

In such a scenario will the spectroscopic mapping show be atomically resolved? Is

it possible to see the atomic lattice while spectroscopic data still averages over large

areas?

A.1.1 Homogeneity and Resolution

To explore this Madhavan and I prepared an STM tip with an irregular but blunt

end. A topograph and it’s fourier transform are shown in figure A.1. Although the

topograph is not perfect the atomic lattice is still evident in the fourier transform.

Within simple models where tip’s end has a hemispherical shape, the atomis periodic-

ity showing in the FT leads one conclude that the tip is one to a few lattice constants

(i.e. in the range of 10-20 A).

To see what spectra look like with this tip and sample a linecut over 3000

73

A were taken, figure A.2. The gap values are clearly homogeneous along these 328

spectra taken every 9 A. A standard gapmap (1282 pixels, 2 meV resolution) on a

640 A FOV was also taken this is part of the first row in figure A.3.

From this data the sample appears to be homogeneous. All of the necessary

characteristics seem to be there: 1) atomic lattice in topography (so the tip is in a

few nanometers at most) and 2) homogeneous spectra over a long distance (5-300 nm

in all). It is important to note that no annealing of the sample was done prior to

cleaving and inserting it into the STM.

To investigate if this could be a product of the tip’s termination geometry

(TTG), we repeatedly battered the tip against the surface to change this configuration

until one piece stuck out more than the others. Topographs with the from the original

and subsequent tips are shown in the first column of Figure A.3. With each TTG a

gapmap was taken. These gapmaps are shown in the second column of Figure A.3.

With each TTG the atomic lattice is evident in the topography, but the variations in

gap distribution change dramatically.

To quantify these variations a histogram was made from the gapmaps in each

row of Figure A.3, shown in Figure A.4. Two systematic changes in the gapmaps are

clear as the TTG was made ”sharper” (i.e. has a more isolated end). The first is

that as the tip becomes smaller (going down the rows in fig. A.3 and from black to

red to blue in fig. A.4) the variation in the gapmap increases. This is because as the

tip gets smaller it is able to measure the local variations in gap. The second trend

is that as the tip gets smaller the average gap measured increases. To explain this

trend one must recall that as the gap value drops the average height of the coherence

peaks increases. This means that by averaging over many spectra the ones with the

lowest gaps will contribute more than those of large gap. So large tips give low and

homogeneous values fore the tunneling gap.

74

3000 Å

Co

nd

uct

ance

(nS)

10

Sample Bias (mV)

-100 -50 0 50 100

Figure A.2: a) A linecut taken with the tip and sample from Figure A.1. It is clear that

the gap changes very little in this tunneling data. Less than %10 over the 350 spectra.

75

640 Å

1.7781.778

20 meV 50 meV50 Å

1

2

3

T(V= -100) ∆ LDOS(V=0)

Row 2

Nu

mb

er

(no

rma

lize

d)

0

1

Fit: ∆0 = 31meV

w = 8 meV

Row 1Fit: ∆0 = 26 meV

w = 3 meV

Nu

mb

er

(no

rma

lize

d)

0

1

Row 3Fit: ∆0 = 33 meV

w = 18 meV

∆ (meV)20 40 60 80

Nu

mb

er

(no

rma

lize

d)

0

1

Figure A.3: Multiple data sets on the same surface with different tips. Each row is a series

of data with a different tip configuration, very blunt, somewhat blunt, and less blunt respec-

tively. Each columns is a different type of data, topography showing “atomic resolution”,

gapmap taken with this tip, and the zero bias conductance showing zero bias resonances.

76

Row 2

∆ (meV)

20 40 60 80

Num

ber

(norm

aliz

ed)

0

1

Fit: ∆0 = 31meV

w = 8 meV

Row 3

∆ (meV)

20 40 60 80

Num

ber

(norm

aliz

ed)

0

1

Fit: ∆0 = 33 meV

w = 18 meV

Row 1

Row 2

Row 3

∆ (meV)

20 40 60 80

Num

ber

(norm

aliz

ed)

0

1

Fit: ∆0 = 26 meV

w = 3 meV

Fit: ∆0 = 31meV

w = 8 meV

Fit: ∆0 = 33 meV

w = 18 meV

Row 1

∆ (meV)

20 40 60 80N

um

ber

(norm

aliz

ed)

0

1

Fit: ∆0 = 26 meV

w = 3 meV

Figure A.4: Three histograms of gap values taken on the same surface on consecutive days

with different TTGs. The Average gap (∆0) and HWHM, taken from a Lorentzian fit are

shown. Clearly as the tip becomes ”sharper” and better able to resolve individual imputiry

resonances the ∆0 becomes greater and w increases dramatically.

77

A.1.2 Necessary Criterion

If periodic structures or gross spectroscopic information are not sufficient to give

one accurate, atomically resolved data on this surface. What is needed? Because

the measured data is the convolution of the data with the tip’s structure, isolated

features with known spatial structure will do the trick. Luckily, Bi2Sr2CaCu208+δ

samples contain such a feature. Impurities that substitute for Cu atoms in the

CuO plane produce strong quasi-bound states that are visible in the low energy

conductance[40, 110, 111, 112, 113, 41]. We have found that even without intentional

doping 1 in 10,000 lattice sites have one of these resonances[41]. We attribute these to

Cu vacancies because of their unique energy (resonance ≈ 0.5 meV) and topographic

characteristics (a depression in the Bi atom above of 0.5 A).

Since they are isolated and aperiodic imaging these structures allows one to get

a detailed image of the tip. To see what this looks like LDOS images taken at V = 0

meV are shown in the third column of Figure A.3. In the first row the only discernable

features is a 250 A repeated shape. This is the shape of the TTG. Although one

can still see the atomic periodicity this tip is 250 A big and averages spectroscopic

information accordingly. Since the gap variation has a correlation length or 30 A it

is no wonder we cannot see any variation in the linecut or associated gapmap since

these vary on 3nm length scale[48, 49, 45]. With the subsequent TTG’s the impurity

resonances in the LDOS(V = 0) are much smaller ≈ 30A, putting the TTG’s on the

order of the gap variation and thus allowing us to resolve the inhomogeneity.

We therefore use impurities as the measure of a tip’s TTG. In all data reported

in this thesis the shape of isolated impurities was used to judge the tip’s acceptability.

In Figure A.5 a gapmap in a large FOV and a conductance map of a single impurity

state is shown for every sample reported on in this thesis.

78

177

180

182

188

c8

c10

Figure A.5: A large FOV gapmap and conductance map of an impurity state from every

surface used for this thesis. The characteristics of each sample are listed in table A.1.

Table A.1: Table showing all data that contributed to this thesis.

Run designation ∆ σ ∆(~r) Tc

177 40 11 640 A 89K OD

180 37 8 655 A 89K OD

182 51 11 530 A 65K UD

188 74K UD

c8 57 15 480 A 55K UD 0.2%Zn

c10 46 10 620 A OPT Ni

79

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