171
c Copyright by Shashank Misra, 2004
c© Copyright by Shashank Misra, 2004
SCANNING TUNNELING MICROSCOPY OF THE
CUPRATE SUPERCONDUCTOR BSCCO
BY
SHASHANK MISRA
B.S., University of Wisconsin - Madison, 1998
DISSERTATION
Submitted in partial fulfillment of the requirements
for the degree of Doctor of Philosophy in Physics
in the Graduate College of the
University of Illinois at Urbana-Champaign, 2004
Urbana, Illinois
SCANNING TUNNELING MICROSCOPY OF THECUPRATE SUPERCONDUCTOR BSCCO
Shashank Misra, Ph.D.Department of Physics
University of Illinois at Urbana-Champaign, 2004Ali Yazdani, Advisor
The cuprate superconductors remain an enigma after nearly 20 years of research. Al-
though most of their properties in the superconducting state can be attributed to d-wave
superconducting order with a short coherence length, very little else about them is under-
stood. In particular, the nature of the electronic state outside the superconducting phase
remains controversial. Whether the anamolous ’normal’ state properties of the cuprate super-
conductors are the result of some remnant of superconductivity or some competing electronic
order, and whether these properties influence superconductivity, remain perhaps the most
important unanswered questions in the field. Because the superconducting phase and many
of the proposed normal phases have variations in their electronic structure on atomic length
scales, a local probe with atomic spatial resolution could be very useful in understanding the
electronic properties of the cuprates.
In this thesis, we will exploit the ability of Scanning Tunneling Microscopy (STM) to
measure the density of states on an atomic length scale to study both the superconduct-
ing and normal state of the high temperature superconductor Bi2Sr2CaCu2O8+δ. In the
superconducting state, we have found a novel signature of d-wave superconductivity- cer-
tain extended defects nucleate a one dimensional zero-energy bound state. The existence of
the bound state can be used as a characterization tool for other defect structures. In the
pseudogap regime, we find an incommensurate, fixed wavelength modulation in the spatial
structure of the density of states that correlates with the pseudogap energy scale. Unlike
similar structures found in the superconducting regime, which are believed to result from
iii
scattering interference, the modulations in the pseudogap regime appear to be some form
of local ordering. Finally, we present the first conclusive identification and density of states
measurements on a single CuO2 plane at the surface of a high temperature superconductor.
Although the strongly correlated electron behaviour of the cuprates is thought to originate
on the CuO2 planes common to these materials, most spectroscopic measurements have
been made on samples terminated with different crystal planes until now. This experiment
demonstrates a different way with which we can access the physics of a doped CuO2 system.
In total, the experiments in this thesis are meant to demonstrate how many of the compelling
mysteries of a highly correlated material are uniquely accessable with an atomic scale probe.
iv
To my fiancee, Julie Shuler.
v
Acknowledgments
It is my pleasure to acknowledge invaluable conversations with Ali Yazdani, Dan Hornbaker,
Michael Vershinin, and Philip Phillips. My group is indebted to the research groups of
Jim Eckstein and Yoichi Ando for providing wonderful high quality samples to work with.
Finally, this work was made possible by support from my parents, Jennifer Tate, Akilan
Palanisami, Richard Hasty, Jonathan Kaufman, Martin Holt, Tim Kidd, and Herve Aubin.
I would also like to acknowledge the following agencies for funding the research presented
in this thesis: NSF through grants DMR-98-75565 and DMR-03-1529632, US-DOE through
grant DEFG-02-91ER4539 via the Frederick Seitz Materials Research Laboratory, and the
ONR through grant N000140110071. All of the research was performed in the Frederick Seitz
Materials Research Laboratory at the University of Illinois. Some basic debugging work was
done in the Center for Microanalysis of Materials, which is partially funded by the US-DOE
under grant DEFG02-91-ER45439. Finally, I would like to thank the University of Illinois
for keeping me fed and clothed my first two years of graduate school through the University
Fellowship and the GAANN Fellowship.
vi
Table of Contents
Chapter Page
1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1
1.1 Why study the high temperature superconductors with scanning tunneling
microscopy? . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1
1.2 Outline . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7
1.3 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9
2 Scanning Tunneling Microscopy . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10
2.1 Theory of Measurement . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10
2.2 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20
3 Scattering in the Superconducting State . . . . . . . . . . . . . . . . . . . . . . . 21
3.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21
3.2 The Andreev Bound State . . . . . . . . . . . . . . . . . . . . . . . . . . . . 26
3.3 Local DOS Modulations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35
3.4 Applications . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 44
3.5 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 46
4 Local Ordering in the Pseudogap Regime . . . . . . . . . . . . . . . . . . . . . . . 50
4.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 50
4.1.1 Precursor Superconductivity Experiments . . . . . . . . . . . . . . . 55
vii
4.1.2 Spin Order Experiments . . . . . . . . . . . . . . . . . . . . . . . . . 58
4.1.3 Theoretical Ideas . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 62
4.2 Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 68
4.3 Scattering Interference . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 73
4.4 Local Order . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 83
4.5 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 86
5 The CuO2 Plane . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 92
5.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 92
5.2 Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 97
5.3 Interpretation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 101
5.4 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 110
6 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 113
Appendix
A STM Methods . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 118
A.1 Construction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 118
A.2 Operation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 124
A.3 Limitations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 129
A.4 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 144
B Elastic Scattering Interference Calculations . . . . . . . . . . . . . . . . . . . . . . 145
B.1 Formulation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 145
B.2 Calculation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 148
B.3 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 160
Curriculum Vitae . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 162
viii
Chapter 1
Introduction
1.1 Why study the high temperature superconductors
with scanning tunneling microscopy?
Some of the most exciting discoveries in condensed matter physics in the last fifty years
have centered on how electrons behave when subjected to strong interactions. In most
materials, the electrons involved in conduction delocalize into a set of plane waves with
a well-defined momentum (wavevector) and energy, and behave like free electrons with a
renormalized mass. Interactions are relatively weak, and can be treated as perturbations to
the free-electron behaviour. However, the ideas we use to understand conventional materials
fail spectacularly in describing materials with strong interactions, which lead to electron-
electron correlations that are responsible for the basic behaviour of the system.
One storied example of a strongly correlated system is superconductivity. [1] The so-
called BCS (Bardeen-Cooper-Schrieffer) superconductors are normal metals for temperatures
above the superconducting transition. However, when these materials are cooled, phonons
mediate an attractive potential between electrons of opposite spin and momenta near the
Fermi energy, and they pair into bosons. This pairing manifests itself as an energy gap in the
1
A CB
Figure 1.1: (A) The High Temperature Superconductors (HTS) are layeredmetal oxides. This schematic drawing shows the positions of the metal ions inBi2Sr2CaCu2O8+δ. (B) Shown here is a schematic diagram of the CuO2 plane,showing the Cu sites (red) and O sites (blue). The doping level of the HTSrefers to the concentration of holes (δ) on each plaquette (green region) in thisplane. (C) The phase diagram of the HTS has two well-established regions:the antiferromagnetic insulator (below TN ≈ 320K and for hole dopings δ ≤0.02), and the superconducting state (below TC ≈ 100K and between δ =.06 − .26). Although its boundaries aren’t well-established, the enigmaticpseudogap regime exists below optimal doping (δ ≈ 0.16), where TC reachesits maximum, and for T ∗ > T > TC .
density of states; simply, you have to pay the energy cost of breaking a so-called Cooper pair
before you can remove an electron from the system. Simultaneously with their formation, the
Cooper pairs condense into a single macroscopic quantum state. This quantum state flows
without dissipation, leading to zero resistance, the first experimental hallmark of supercon-
ductivity. Moreover, the state has a macroscopic phase, with the Cooper pairs remaining
coherent over long length scales. A non-trivial consequence of this is the expulsion of exter-
nally applied magnetic fields, the second major experimental hallmark of superconductivity.
The theoretical explanation of superconductivity, BCS theory, is one of the landmark devel-
opments in the field of condensed matter physics: not only did it completely describe the
effect, it demonstrated that the solution to strongly correlated problems lies in determining
the novel states that correlations can force electrons into. [2]
The discovery of High Temperature Superconductors (HTS) understandably came as a
2
shock, as their superconducting transition temperatures are significantly higher than pre-
dicted by the BCS theory. [3] However, the HTS are more than simply high transition
temperature analogues of their low temperature counterparts. [4] Instead of being normal
three dimensional metals, the parent compounds of the HTS are layered metal oxides, with
superconductivity occuring in dopable CuO2 planes common to these materials (Fig. 1.1a).
In a phenomenological sense, we understand the strongly correlated phases at the extremes
of the doping-temperature phase diagram fairly well- the undoped insulator appears to be
a Heisenberg antiferromagnet [5], and the heavily overdoped superconductor appears to be
a quasi-two-dimensional metallic BCS superconductor (Fig. 1.1b). [4] However, stemming
from the poor understand of the microscopic details of either these phases, we have a poor
understanding of the strongly correlated behavior in between the two extremes. [6] The HTS
have become the poster child for the study of strongly correlated problems, as many of the
issues in understanding the middle of the cuprate phase diagram are general.
In their undoped state, the traditional theory of metals would predict that the CuO2
plane would have one half-filled band of states at the Fermi energy. However, there is a
strong Coulomb repulsion between these electrons (one electron per unit cell), and they
localize on the Cu sites, forming an insulator. The system lowers the total energy of this
configuration by forcing neighboring sites to have anti-aligned spins. Thus the undoped,
parent state of the CuO2 plane itself has strong correlations which lead to long-range anti-
ferromagnetic order. Adding holes (or removing electrons) from the CuO2 plane suppresses
this ordering much more quickly than would be expected from the simple dilution of the spin
lattice (Fig. 1.2a). [7] The phenomenological reason for this is relatively simple- the addition
of a very small number of holes (.02 per unit cell) introduces itinerant states at the Fermi
level. [8] While ordering in the undoped compound is well-understood theoretically, we have
a poor understanding of the relationship between short-range (10 − 100 A) in-plane anti-
ferromagnetic correlations, which survive the suppression of global antiferromagnetic order,
3
A BA
⟨π,π⟩
⟨0,π⟩
Figure 1.2: (A) This is a schematic diagram of the CuO2 plane at half-filling (adoping of δ = 0). The removal of a single spin on this 5×5 grid, correspondingto a doping of δ = .04, is enough to destroy antiferromagnetic order in theHTS. (B) A schematic diagram of the amplitude (radial distance) and phase(sign) of the d-wave order parameter of the HTS as a function of angulardirection. Here, 〈0, π〉 refers to the Cu-O bond direction of the CuO2 plane,and 〈π, π〉 is 45 rotated from the bond direction.
and these electronic states. [5] This problem is generic to the study of strongly correlated
systems- it remains unclear what happens when electron kinetic energy and correlation ener-
gies are of similar magnitude. What is clear is that these electronic states strongly interact
with some mode of the system, and will superconduct at sufficiently low temperatures with
the addition of only a few more holes.
Superconductivity sets in at sufficiently low temperatures upon the addition of between
.06 and .26 holes per CuO2 unit cell. This superconducting order is understood to be
BCS-like with two significant modifications of the BCS theory. The superconducting wave
function has a phase that changes sign upon π/2 rotations, with a corresponding node in
the superconducting gap between oppositely signed lobes of the order parameter (Fig. 1.2b).
[9] The coherence length for superconducting correlations is also many orders of magnitude
smaller than in BCS superconductors– just 16A in-plane and 3A out-of-plane. [10] A modified
BCS theory that includes these details has proven remarkably successful at explaining most
experimental observations in the superconducting state. [4] However, key issues, like the
reason why the superconducting phase forms a dome in the phase diagram, remain open
4
questions. For overdoped samples, the energy gap in the density of states is seen to shrink,
which should result in falling TC values. However, for underdoped samples, the value of the
energy gap is increasing while TC is falling, suggesting that another energy scale is involved
in the formation of the superconducting state. The appearance of two energy scales that
separately influence the ground state of a system is, too, a recurrent theme in the study of
strongly correlated electron systems.
The importance of this second energy scale is magnified by the curious properties of
the state above the underdoped half of the superconducting dome- the so-called pseudogap
regime. [6] The energy gap in the density of states evolves smoothly through the super-
conducting transition, with the gap filling in with states upon reaching a temperature T ∗.
Because the pseudogap is symmetric about the Fermi energy, it intuitively appears to result
from correlation effects, which might determine the electronic phase of the system. One
prominent proposal posits that fluctuating superconducting correlations survive the global
phase transition. The pseudogap is then a signature of this fluctuating superconductivity,
where global phase coherence has been lost, but Cooper pairs still exist. The other energy
scale is then associated with phase coherence of the Cooper pairs. The shape of the super-
conducting dome results from this energy scale shrinking in the underdoped regime, and the
pairing energy scale shrinking in the overdoped regime. The idea that a mysterious energy
scale might be controlling global phase coherence is not as far-fetched as it sounds. In purely
two dimensional systems, off-diagonal long-range order should set in only at T = 0. However,
most real systems are quasi- two dimensional, and off-diagonal long-range order sets in below
a temperature related to the anisotropic nature of the interactions in the sample, like the
energy scale for out-of-plane interactions. Meanwhile, vestiges of local order can survive up
to much higher temperatures, associated with the energy scale for in-plane interactions. Such
effects are common in planar correlated systems, and are present in the antiferromagnetic
insulating regime of the HTS. [7]
5
Another set of proposals posits that the pseudogap results from some exotic order, such
as charge and/or spin order. In the HTS, given the complex interplay of charge and spin
degrees of freedom and the proximity to the antiferromagnetic insulator, the most obvious
form of ordering would be one derived from residual antiferromagnetic correlations. Any
number of ordered states have been proposed for the pseudogap regime, some competing
with superconducting order, and others either facilitating or remaining indifferent to it. [4]
Most of these proposals, however, involve modulations of either spin or charge degrees of
freedom on atomic length scales. The energy gap observed in the underdoped part of the
phase diagram is then associated with this ordering, and the superconducting gap masks
the energy gap associated with this order for overdoped samples. Finally, the shape of the
superconducting dome is then determined solely by the energy gap associated with Cooper
pairing. The idea of competing order parameters, or of two different kinds of order being the
manifestation of a multi-dimensional order parameter, is also a popular concept in describing
many correlated electron systems.
After more than fifteen years of investigation, significant questions remain unanswered
about the cuprate superconductors, clearly requiring new experimental tools to elucidate
them. Because the relevant length scales throughout the cuprate phase diagram are of
the order of tens of Angstroms, Scanning Tunneling Microscopy (STM) is one of the most
promising such tools. STM allows us to measure both topographic features, allowing us
to get a snapshot of the local surface structure, and spectroscopic features, allowing us to
determine the local density of states, each with sub-Angstrom spatial resolution. In this
thesis, we will investigate the electronic density of states on the atomic scale in each of the
regions of the phase diagram. We hope to demonstrate how this atomic-scale view leads
to both new basic information about the cuprates, and tools which can be used in future
research.
6
1.2 Outline
The superconducting state of the HTS is well-described as a d-wave BCS superconductor with
a short coherence length. In chapter 3 of this thesis, we will examine a novel consequence of
both these characteristics. When the superconducting state scatters strongly off structural
defects, a novel one-dimensional state at zero energy forms within a coherence length of the
defect as a consequence of the d-wave phase of the pair potential. Because the zero bias state
should only appear near structures that strongly scatter superconducting quasiparticles, its
appearance can be used as a test of the strength of scattering structures about which little is
known. We will also examine the spatially-resolved density of states far from these structures.
We find that the in-gap electronic states form dispersing standing waves in real space. Many
of their characteristics are shown to be consistent with weak scattering, which is sensitive to
the amplitude of the pair potential. Although neither result is a new discovery by itself, they
each have useful application for the study of the cuprate superconductors. First, the zero-
bias state is a novel consequence of having d-wave superconducting correlations. It can thus
be used to identify whether or not a sample has superconducting correlations using density
of states measurements, which are a single-particle probe, even when the phase of the sample
is unknown. Second, the Born formalism used to describe the modulated patterns can be
used to model any modulated density of states pattern. In situations where the electronic
structure of a sample is unknown, how well different candidate electronic states match the
measured modulated patterns could be used to distinguish between the candidate states.
The pseudogap in the density of states, on the other hand, remains an enigma after a
decade of intense research into its origin. Whether it results from residual superconducting
correlations, from some other form of ordering, or some other phenomenon entirely remains
an open question. In chapter 4 of this thesis, we will describe how we find that the electronic
states inside the pseudogap form a static, incommensurate standing wave pattern. While
7
similar patterns have been found in samples below TC , the patterns found in the pseudogap
regime are unique in that their wavelength is fixed as a function of energy. We find that the
modulations cannot be described by simple Born scattering, which suggests that the patterns
in the pseudogap state result from local ordering. Furthermore, it now appears that when-
ever superconductivity is suppressed, either by decreasing doping, increasing temperature or
applying a magnetic field, local electronic order develops in the density of states.
All the correlated electronic behaviour of the cuprates, from superconductivity to the
pseudogap to the antiferromagnetic insulator, originate on the CuO2 planes common to
these materials. However, electronic density of states measurements have typically been made
using surface sensitive techniques on samples terminated by different crystallographic planes.
In Chapter 5 of this thesis, we will describe how we have made the first definitive density
of states measurements on a single CuO2 plane at the surface of a cuprate superconductor.
The unexpected density of states measured on samples terminated by the CuO2 plane will
require us to consider the possibility that exposing this plane at the surface has changed
its doping level, and hence its phase. Although more measurements, such as searching for
zero-bias states and looking for spatial structures in spatial maps of the density of states,
will be required before we can identify its electronic phase, this discovery opens up a new
avenue with which to examine cuprate physics.
The main result of this work is to demonstrate that Scanning Tunneling Microscopy, with
its ability to correlate local structural and electronic information, is capable of revealing
information about the cuprate superconductors that is simply inaccessible to other probes.
In general, using methods similar to the ones used in this thesis, STM appears poised to
make significant contributions to the study of any correlated electronic system with short
characteristic length scales.
8
1.3 References
[1] J. Schrieffer and M. Tinkham, Rev. Mod. Phys. 71, 313 (1999).
[2] J. Bardeen, L. Cooper, and J. Schrieffer, Phys. Rev. 108, 1175 (1957).
[3] J. Bednorz and K. Muller, in Nobel Lectures, Physics 1981-1990, edited by T. Frangsmyr
and G. Ekspang (World Scientific Publishing, ADDRESS, 1987), Chap. Perovskite-Type
Oxides- The New Approach to High-Tc Superconductivity.
[4] M. Norman and C. Pepin, Rep. Prog. Phys. 66, 1547 (2003).
[5] M. Kastner, R. Birgeneau, G. Shirane, and Y. Endoh, Rev. Mod. Phys. 70, 897 (1998).
[6] T. Timusk and B. Statt, Rep. Prog. Phys. 62, 61 (1999).
[7] B. Keimer et al., Phys. Rev. B 46, 14034 (1992).
[8] T. Yoshida et al., Phys. Rev. Lett. 91, 27001 (2003).
[9] D. V. Harlingen, Rev. Mod. Phys. 67, 515 (1995).
[10] U. Welp et al., Phys. Rev. Lett. 62, 1908 (1989).
9
Chapter 2
Scanning Tunneling Microscopy
2.1 Theory of Measurement
Many tools already exist that allow scientists to measure surface structure, and the electronic
properties of these surfaces. Over the last two decades, scanning electron microscopy has
become a workhorse in measuring local surface structure, and angle-resolved photoemission
has become the de−facto standard for measuring the electronic density of states as a function
of energy and momentum. What makes Scanning Tunneling Microscopy unique is its ability
to measure the density of states locally with sub-Angstrom resolution, and then to correlate
these measurements with simultaneously-acquired topographic information. Although the
methods discussed in this chapter have originated from the study of normal metals, they
offer the tantalizing ability to investigate short-length-scale correlations in any number of
systems, which we will demonstrate explicitly for the HTS in the rest of this thesis. A more
complete discussion of the experimental methods and sources of noise appears in Appendix
A.
In an STM experiment, a sharp metallic wire is brought close enough to the surface of a
sample that electrons can tunnel between them (Fig. 2.1a). [1] If the sample is biased at a
10
voltage V with respect to the tip, the Golden Rule dictates that a net tunneling current
I(~r, V ) =4πe
h
∞∫
−∞
dE|T 2(d, E, E − V )| ×
[ρT (E)f(E)][ρS(~r, E − V )(1 − f(E − V ))] −
[ρT (E)(1 − f(E))][ρS(~r, E − V )(f(E − V ))] (2.1)
flows from the tip to the sample [2, 3, 4], where T (d, E, U) is the tunneling matrix element for
an electron between the states at energy E in the tip and the states at energy U in the sample,
d is the tip-sample separation, ρT (E) is the density of electronic states (DOS) in the tip at
an energy E, ρS(~r, E) is the DOS of the sample at a position ~r and an energy E, and f(E)
is the Fermi function. Two reasonable assumptions simplify Equation 2.1 into something
more intuitively understandable. First, the tip used in most STM experiments is a simple
metal with a flat density of states over the energies explored by the experiment, allowing us
to set ρT (E)=ρT (0). Second, we will assume, for now, that none of the transitions between
states in the tip and states in the sample are more favorable than others. The tunneling
matrix element will then be given by the overlap of the tip and sample wavefunctions. If
the energy height of the barrier, given by the average of the two work functions (typically
3 − 5eV ), is significantly greater than the energy curvature of the barrier (given by the
difference of the two work functions added to the sample bias, typically 0 − 1eV ), the
tunneling matrix element can be treated with a simple WKB approximation. This gives
T (d, U, E) = T (d) = exp(−d√
8mφh2 ), where m is the mass of an electron, and φ is the average
of the tip and sample work functions. The simplified expression,
I(~r, V ) =4πρT (0)e
he−d
√
8mφ
h2 ∞∫
−∞
dEρS(~r, E)[f(E + V ) − f(E)], (2.2)
reveals that the the tunneling current changes proportionally to the integral of the local
density of states of the sample between EF and EF + V as a function of voltage, and
exponentially as a function of the tip-sample separation.
11
Ψ
Ψ
Tip
Sample
overlap
region
Tip
VI
Sample
Vacuum
Energy EF
EF
+V
~Φ
De
nsi
ty o
f S
tate
s (x(t),y(t))Z(t)
I
Itunnel
ref-A B
Figure 2.1: (A) (left) This is a schematic diagram of an STM junction. Theexponential dependence of tunneling current on tip-sample separation derrivesfrom the overlap of the evanescent wavefunctions of the tip and sample in thevacuum barrier. (right) Applying a voltage bias to the tip relative to thesample yields a net tunnel current flowing from the tip to the sample (sampleto the tip) proportional to the number of occupied states in the tip(sample)convoluted with the number of empty states in the sample (tip). (B) A feed-back loop ensures that the tunnel current remains constant by controlling thetip-sample separation z(t). Recording z(t) while rastering the tip across thesurface (x(t),y(t)) yields a height-field of the surface that is called a topograph.
A number of theorists, most prominently Harrison, [5] have objected to this formulation
of the tunneling current. They point out that the tunneling matrix element should be a tun-
nelling attempt frequency multiplied by the probability for successful electron transmission
through the potential barrier, with the former having been omitted by the above treatment.
This attempt frequency can be estimated as the particle flux incident on the tunnel barrier,
which, from dimensional arguments, is inversely proportional to the velocity of the parti-
cles. In a three-dimensional metal, the attempt frequency cancels out the density of states
term in the tunneling current. However, for the materials investigated here, which cannot
be approximated as simple three dimensional metals, we do not expect such a cancellation
to occur. Moreover, as demonstrated by the pioneering work of Giaever [6], tunneling does
sensitively probe the density of states in a superconductor. Consequently, we proceed with
the formulation presented in Equation 2.2 above.
The tunneling current’s exponential dependance on tip-sample separation can be ex-
12
A
B
C
D
Figure 2.2: These are STM topographs of the same 700A × 350A area of aAu(111) surface taken under two different tunneling conditions: I = 200pAand V = +100mV for (A) and I = 200pA and V = −100mV for (B). Tohighlight the modulated pattern, we have taken a spatial derrivative of (A),shown in (C), and of (B), shown in (D). The location of the crests of themodulated pattern have been designated by the yellow lines in (C). Plottedin (D) are the same lines, which no longer match up with the crests of themodulations.
ploited to learn about the topography of the surface (Fig. 2.1b). [7] To take a topographic
image of the surface, a feedback loop is used to keep the tunneling current constant while
the tip is rastered across the surface. The output of the feedback loop, which changes the
height of the tip in response to changes in the tunneling current, is recorded as a function of
position to produce an image. If the sample has the same integrated density of states every-
where on the surface, the feedback loop will change the height of the tip solely in response
to structural changes at the surface. In this way, heights of features such as step edges can
be exactly determined. However, the density of states is almost never homogeneous across
a sample, and the feedback loop also changes the tip’s height in response to these changes.
Thus topographic images contain a convolution of both structural and electronic informa-
tion, and must be considered contours of constant electron density, not merely local maps of
surface structure.
13
The canonical examples of this convolution of structural and electronic information are
topographic images taken on certain surfaces of noble metals, such as Au(111) (Fig. 2.2). The
STM topographs of the Au(111) surface contain three main features: line defects separating
two terraces of different height [7], a herringbone feature [8], and a modulated pattern found
near the line defects. [9, 10] The structural nature of the line defect can be confirmed by
taking topographs at different voltages. Changing the voltage should change the integrated
density of states, with the feedback loop responding to this change in the tunneling current
by changing the tip height. If the line defect is a structural feature, the height of the two
terraces should be the same in both topographs. Averaging all the lines in each topograph
together to remove the effects of the other two topographic features, the height separating
the two terraces is found to be equal (2.4A), proving that the line defect is a structural
feature. What’s more, it agrees precisely with the distance between two adjacent atomic
planes of the crystal, indicating that the step edges are each one-atom high. The herringbone
pattern also does not change significantly between the two topographs, and hence can also
be identified as having a structural origin. The modulated pattern, on the other hand, has
a different period of modulation for the two biases (Fig. 2.2c & d), indicating its origin is
primarily electronic. Spectroscopic information, which can be acquired simultaneously with
topographic information, can be used to further elucidate the origin of these features.
Scanning tunneling microscopy can also be used to measure the local density of states.
Starting from Equation 2.2, the differential conductance of the tunnel junction is given by
dI(~r, V )
dV=
4πe|T 2(d)|ρT
h
∞∫
−∞
dEρS(~r, E)d
dV[f(E + V )]. (2.3)
In the scheme discussed here, positive voltages correspond to energies above the Fermi energy
(empty states), and vice versa. At zero temperature, the differential conductance as a
function of voltage is then directly proportional to (only) the sample density of electronic
states at energies relative to the Fermi energy. [3, 4] This result has been confirmed by
Yazdani and coworkers, who demonstrated that the conductance as a function of bias voltage
14
A
B C
D E
F
Figure 2.3: (A) This is an 560A × 560A STM topograph of Au(111) takenwith a I = 100pA and V = 200mV junction at T = 100K. Conductancemaps were acquired simultaneously with this topograph at a range of voltagesusing lockin parameters f = 513Hz, dV = 3mVrms, and τ = 20mS. Shownare subsections of these conductance maps taken in the square region in (A)at (B) V = −415mV , (C) V = −288mV , (D) V = −161mV , and (E) V =−34mV . Also shown is the conductance map of the entire region in (A) takenat V = 93mV (F).
on Nb(110), a BCS superconductor, exactly matches the BCS density of states. [11] This
spectroscopic ability can be used to examine the spatial arrangement of the density of states
at a given energy, or the energy-dependence of the density of states at a given position. To
take either measurement, the tip is moved to a particular location on the sample with the
feedback loop closed. The feedback loop is then opened, the bias is changed to the voltages
of interest, and the differential conductance of the tunnel junction is recorded. The feedback
loop is then closed, and the tip is moved to the next point where density of states information
will be measured. This method of taking the data introduces the (unavoidable) complication
that the feedback loop, in an attempt to keep the tunneling current constant, is normalizing
the integrated density of states at every point that spectroscopy data is taken. However,
if the local variation of the density of states does not result in features that show up in
topographic images, this normalization has a minimal effect.
15
A D
0.00
0.10
0.20
0.30
0.40
-400 -300 -200 -100 0 100 200
Wav
en
um
be
r (1
/Å)
Energy
CB
Figure 2.4: (A & B)These are false-color images of the Fourier amplitudeof the data in Fig. 2.3c&e. The white arrows point at a dispersing featurecorresponding to the standing wave pattern seen near the step edge. (C)The wavevector can be determined for a set of energies, mapping a dispersionrelation for the modulated patterns (closed circles). The dispersion of anelastic scattering interference model based on photoemission data, as discussedin the text, is shown as open circles. (D) ARPES data from Ref. [12] clearlyresolves the dispersion of a surface state. Note that ARPES has resolved asplitting of the dispersion. The width of the peaks in Fourier transforms of theSTM data (≈ .06A−1) are approximately equal to this splitting, and precludeit from being resolved in the STM data.
Returning to the simple example of the Au(111) surface, local density of states maps can
be used to characterize the modulated electronic pattern seen in topographs next to step
edges. From topographs taken at different voltages, the modulations were hypothesized to be
electronic in origin. The DOS maps confirm this- the modulated pattern next to step edges
has a wavelength that changes with energy (Fig. 2.3). [10] To more thoroughly characterize
the wavelength of these periodic modulations, we can take Fourier transforms of the DOS
maps (FTDOS, Fig. 2.4a&b). [13] The FTDOS maps taken at several energies can then be
used to compile a dispersion relation for this feature (Fig. 2.4c). ARPES data on Au(111)
shows that a two-dimensional surface band is located at these energies (Fig. 2.4d), but
at a wavevector exactly half that seen by STM. The two sets of data are connected by a
phenomenon known as elastic scattering interference.
In picture terms, there are two roughly equivalent ways to understand elastic scattering-
16
q
k=(0,0)
k=(π,π)
k=(0,π)
k=(0,0)
k=(π,π)
k=(0,π)q
q=(0,0)
q=(π,π)
q=(0,π)
A CB
Figure 2.5: Here we illustrate both the wave-based and particle-based toymodels for scattering interference. (A) In a wave-based picture, electron wave
functions at a given energy have a set of resitricted ~k values (blue circle). Here,the thin black square extends between (±π,±π). Density of states modulations
resulting from interaction with defects have a wavevector ~q = 2 × ~k (redcircle). (B) Alternatively, we can take a particle-based view. Electrons existon contours of constant electron energy, which is the same blue circle as in (A).Density of states modulations are expected for the set of wavevectors ~q joiningthe points on these curves (red). The largest density of states contributionswill be for the wavevectors connecting parallel parts of the curves of constantelectron energy. (C) Compiling the full set of these wavevectors ~q at a givenenergy yields a scattering map which agrees with the wave-based picture in(A).
17
a wave-based picture [13] (Fig. 2.5a) and a particle-based picture [14, 15] (Fig. 2.5b). The
density of states over regions with no scattering centers contains electron wave functions
with many different momenta (~k) and phases incoherently superposed with one another. A
defect can pin the phase of electron wave functions at the scattering center, making incoming
and outgoing waves interfere with one another to form standing waves. Electrons at a given
energy have a set of allowed ~k vectors; at the Fermi energy, the locus of allowed ~k vectors is
the Fermi surface. The electron density, however, is given by the square of the wavefunction,
and thus has a wavevector twice as long as the wavevector of the wavefunction. The density
of states patterns will then occur for a set of wavevectors ~q = 2 × ~k. Equivalently, we can
consider scattering in particle terms. Electrons at a given energy exist on curves of constant
electron energy; at the Fermi energy, this curve is the Fermi surface. An electron with a
given ~ki from this set can scatter elastically off a defect to a different wavevector from this
set ~kf . The density of states correction from scattering will occur for the set of wavevectors
~q = ~kf − ~ki. As can be seen in Figure 2.5a&c, the wavevectors comprising the density of
states correction from elastic scattering produced by these two pictures is identical. Applying
either these two procedures to the ARPES data from Au(111), we can calculate a dispersion
curve that agrees with the dispersion of the standing wave pattern seen near step edges using
STM (Fig. 2.4c). The modulated patterns seen in STM can thus be understood to result
from the elastic scattering of the Au(111) surface state off the step edge.
The herringbone pattern, which we suspected to be primarily structural in origin from
the STM topographs, also has an electronic component (Fig. 2.6). [16] Taking conductance
measurements as a function of energy both in the small flat region between two herringbone
corrugations and between two pairs of herringbone corrugations shows that there is an ad-
ditional contribution to the density of states at the herringbone centered around −425meV .
At this energy, elastic scattering of the surface band has a real-space period of ≈ 63A, which
is precisely the spacing between pairs of the herringbone corrugations. The herringbone pat-
18
3
4
5
6
7
8
9
-0.6 -0.4 -0.2 0 0.2 0.4 0.6
Spectra on Au 111 surface
hcp regionherringbone crestfcc region
dI/
dV
(a
rb.
un
its)
Energy (V)
fcc regionhcp region
herringbone crestA B
Figure 2.6: (A) An STM topograph of a 270A × 270A region (I = 130pAand V = −100mV ) rendered in 3D. The long flat regions (38A) betweenherringbones has an fcc structure, and the short region (25A) in between thetwo ridges has an hcp structure. [8] (B) Taking point spectra on top of anfcc region, an hcp region, and the ridge separating the two (the crest of theherringbone pattern) reveals an increased number of states accumulate in thehcp region at about −425mV and an increased number of states accumulateat the crests at about −375mV and −275mV . Data were taken at T = 80K.
tern is effectively acting like a Fabry-Perot resonator for the surface state. As a further test
of this hypothesis, spectra were also taken on the crest of the herringbone corrugation, which
revealed an additional contribution to the density of states at both −375mV and −275mV .
At these energies, elastic scattering of the surface band should produce a real-space period of
38A and 25A, respectively. These two periods are exactly the size of the the large flat region
between two sets of crests and the small flat region between two crests. [8] Although the data
set isn’t conclusive, it appears that the herringbone pattern acts like a weak scattering cen-
ter for the surface state band, with an enhanced density of states at energies (wavelengths)
where the surface state can resonate between parts of the herringbone structure.
19
2.2 References
[1] G. Binnig and H. Rohrer, in Nobel Lectures, Physics 1981-1990, edited by T. Frangsmyr
and G. Ekspang (World Scientific Publishing, ADDRESS, 1993), Chap. The Develop-
ment of the Electron Microscope and of Electron Microscopy.
[2] J. Bardeen, Phys. Rev. Lett. 6, 57 (1961).
[3] J. Tersoff and D. Hamann, Phys. Rev. Lett. 50, 6858 (1983).
[4] J. Tersoff and D. Hamann, Phys. Rev. B 31, 805 (1985).
[5] W. Harrison, Solid State Theory (McGraw-Hill, ADDRESS, 1970).
[6] I. Giaever, Phys. Rev. Lett. 5, 147 (1960).
[7] G. B. (sic), H. Rohrer, C. Gerber, and E. Weibel, Phys. Rev. Lett. 49, 57 (1982).
[8] J. Barth, H. Brune, G. Ertl, and R. Behm, Phys. Rev. B 42, 9307 (1990).
[9] M. Crommie, C. Lutz, and D. Eigler, Nature 363, 524 (1993).
[10] Y. Hasegawa and P. Avouris, Phys. Rev. Lett. 71, 1071 (1993).
[11] A. Yazdani et al., Science 275, 1767 (1997).
[12] M. Hoesch et al., Phys. Rev. B 69, 241401 (2004).
[13] L. Peterson et al., Phys. Rev. B 57, 6858 (1998).
[14] J. Hoffman et al., Science 297, 1148 (2002).
[15] K. McElroy et al., Nature 422, 592 (2003).
[16] W. Chen, V. Madhavan, T. Jamneala, and M. Crommie, Phys. Rev. Lett. 80, 1469
(1998).
20
Chapter 3
Scattering in the Superconducting
State
3.1 Introduction
The superconducting state of the HTS is referred to as being unconventional since, although
a modified BCS theory has been remarkably successful in describing their properties, their
measured properties are often fundamentally different than their BCS counterparts. Per-
haps the most significant differences arise from the pair potential in HTS having a d-wave
symmetry, compared with s-wave symmetry for BCS superconductors (Fig. 3.1a). This un-
conventional pairing symmetry directly impacts most measurements through the density of
states, which is sensitive to the amplitude of the pairing potential. In the HTS, the pairing
potential has nodes arising from its d-wave nature. Averaging the density of states equally
for all angles produces a V-shaped energy gap with in-gap states, in contrast to the s-wave
case, which has no in-gap states (Fig. 3.1b). More spectacular differences between s and
d-wave superconductors arise in experiments sensitive to the phase of the pairing potential.
The symmetry of the pair potential was originally established in experiments probing
21
-2 20 -2 20
1
2
1
2
Ε/∆ Ε/∆
Ν (
Ε)/
Ν (
Ε )
s
0
F
Ν (
Ε)/
Ν (
Ε )
s
0
F
<0,π>
<π,π>
<0,π>
<π,π>
A B
C D
++
-
-
Figure 3.1: The s-wave pair potential is isotropic as a function of angle (A),while the d-wave pair potential changes sign upon π/2 rotations (B). Theamplitude of the pair potential affects many measurements through the densityof states, shown here for both the s-wave (C) and d-wave (D) pair potentials.The latter was calculated by averaging the density of states equally for allangles.
the phase of the superconducting order parameter, which has the same phase as the pair
potential. These experiments found novel electronic effects arising from the sign change of
the d-wave order parameter upon π/2 rotation, which simply don’t exist for isotropic s-wave
order parameters. [1, 2] In one of these, a superconducting loop was formed by joining
orthogonal edges of a superconducting HTS sample with a BCS superconductor (Fig. 3.2).
The single quantum state that the phase-coherent Cooper pairs have condensed into can
be described by a pseudo-wavefunction that, thermodynamically, is the order parameter for
superconductivity. The square of the amplitude of this wave function is the density of Cooper
pairs in the condensed state, and the phase of the condensate has the symmetry of the pairing
potential. When two superconductors are connected to one another, a current flows between
them in response to any change in the phase of the superconducting order parameter across
22
A B C
Figure 3.2: The dc-SQUID experiment was used to show that the order pa-rameter in HTS changes sign upon rotation by π/2. A square block of testsuperconductor is connected into a loop by an isotropic s-wave BCS super-conductor. If the phase of the test sample does not change sign at the twojunctions, the currents from the two junctions will cancel one another (A andB), and there is no net current around the loop. However, a corner junction fora d-wave superconductor has an additive contribution from the two junctionsbecause the order parameter changes sign on π/2 rotation. (Figures adaptedfrom Ref. [1]).
the junction (the dc Josephson effect), even in the absence of an applied field. If HTS had
an order parameter that did not change sign upon π/2 rotation, the current flow from one
junction of the corner SQUID would cancel the current flow from the other. In fact, there is
a net current flow around the loop, confirming that that the order parameter does not have
s-wave symmetry. Further details in this work showed that the order parameter not only
changes phase on π/2 rotations, but has a strictly d-wave character.
Potential scattering is another sensitive probe of the d-wave pair potential, and produces
novel phenomena which have no analog for s-wave superconductors. [3] In an s-wave su-
perconductor, there are no available single-particle states at low energies for potentials to
scatter unless they break a Cooper pair apart. Consequently, potentials that do not break
time-reversal symmetry, like magnetic impurities, have no effect, invariant of the strength
23
of scattering. [4] In a d-wave superconductor, on the other hand, there is a finite density
of low-energy quasiparticle states because the magnitude of the pair potential goes to zero
at certain angles. [5, 6] For strong potentials, including non-magnetic impurities [7, 8, 9],
surfaces, and interfaces [10, 11, 12, 13, 3], the scattering of these low-energy states can lead
to the suppression of unconventional order parameters. [14, 15] Quasiparticles inside this
region of suppressed superconductivity can scatter in one of four ways off the pair potential
of the superconducting region (Fig. 3.3a). Two of these are intuitively obvious- reflection or
transmission of the injected quasiparticle. Because BCS superconductivity involves a mixing
of electron-like and hole-like quasiparticles, a third possibility is that the injected quasiparti-
cle is transmitted through the boundary as a quasiparticle of the opposite type. Finally, the
most interesting of these is the so-called Andreev process, in which a quasiparticle injected
into the superconductor pairs with another quasiparticle in the superconductor to form a
Cooper pair, and a quasiparticle of the opposite charge is retro-reflected from the interface.
This retro-reflected quasiparticle then reflects off the strong scattering potential, and can
subsequently be Andreev scattered again by the pair potential of the superconductor (Fig.
3.3b). In superconductors with higher pairing symmetries (non s-wave), the pair potential
that scatters the quasiparticle can be different for the two Andreev processes. In the case
that the lobes of the d-wave order parameter are oriented at a π/4 angle to the scatter-
ing interface, the pair potential has opposite sign (and the same magnitude) for successive
Andreev reflections for all possible quasiparticle trajectories. As a direct consequence of un-
conventional pair potential symmetry, the back-scattered quasiparticles interfere coherently
to form in-gap states that are bound to the scatterer.
This so-called Andreev bound state, which is a novel signature of the sign change in
the pair potential, should appear as a peak in the in-gap single particle density of states.
Indeed, a zero bias conductance peak has been seen in a large number of both planar and
STM tunnel junctions on 110 surfaces, a geometry that should produce an Andreev bound
24
injected quasiparticle
reflected quasiparticle
1
injected quasiparticle Electron-like quasiparticle2
Normal Superconducting
Hole-like quasiparticle3 injected quasiparticle
4 injected quasiparticle
Hole-like quasiparticle
Cooper Pair
A B
++
-
-
++
-
-
1. Injected electro
n
2. Retro-re
flected hole
3. Reflected hole
4. Retro-reflected electron
Normal Superconducting
θ
sc
att
eri
ng
in
terf
ac
e
Figure 3.3: (A) A quasiparticle in a region of normal material can scatter inone of four ways off the pair potential of a superconductor. (B) The region ofsuppressed superconductivity (normal) can support quasiparticles (1) whichundergo Andreev retro-reflection from the normal-superconductor interface(2), are then reflected by the scattering interface (3), and are finally retro-reflected again by at the normal-superconducting interface (4) in a closed loop.
state at the Fermi energy (Fig. 3.3c, with θ = π/4). [16, 17, 18, 19, 20, 21, 22, 23, 24, 25]
Although the in-gap states seen in these experiments could arise from other sources, the
accumulated evidence [3], particularly the state’s sensitivity to disorder [26, 27], suggests
that the effect originates from Andreev Scattering. Still, to date, a direct observation of
neither the suppression of the order parameter near such an interface, nor of the spatial
confinement of the state to within a coherence length of the interface, has been made. STM
investigations of the scattering from both non-magnetic [28, 29, 30, 31] and magnetic [32]
pointlike impurities have also shown in-gap resonances in the density of states within a
coherence length of the scattering center. Although the theoretical understanding of these
in-gap resonant states has largely been in the context of either Born or unitary scattering
[7, 8, 9], their presence has also been posited to be an extention of the Andreev mechanism.
[33]
In this chapter, we will extend both bodies of work by measuring the local density of states
25
both near and far away from spatially extended defects. [34] We find that step edges running
along the 110 direction strongly suppress superconducting order- the superconducting gap
in the density of states fills in upon approaching the step. Furthermore, we find a novel
zero-bias state that exists only within a coherence length of the step, and extends for several
coherence lengths along it. This results from Andreev scattering at the step edge, which
is sensitive to the phase of the d-wave pair potential. This behavior is in contrast to our
finding that 110 twin boundaries show no suppression of the order parameter, nor any
bound state. This suggests, contrary to published theory [35, 36], that twin boundaries are
not strong scatterers, and may result from a more gradual atomic-scale strain field. [37]
Finally, the low energy electronic states far away from defect structures are seen to form
spatially modulated patterns in real space. [38, 39, 40, 41] Although no scattering center
can be identified in the data, it has been suggested that these patterns result from weak
scattering, which probes the amplitude of the pair potential. [42, 43, 44, 45, 46, 47] We
conclude by proposing how the results presented in this chapter can be applied to situations
where the electronic ground state of the system is not well-understood.
3.2 The Andreev Bound State
To examine the effects of spatially extended scattering structures in the HTS, we performed
STM measurements on the superconducting state of 1000A thick Bi2Sr2CaCu2O8+δ thin
films grown using molecular beam epitaxy. The slightly underdoped samples had a resistive
superconducting transition at 84K and a transition width of 4K. The samples were mechan-
ically cleaved along the c-axis direction in UHV at room temperature prior to insertion into
a home-built UHV STM operating at liquid helium temperatures (LTSTM). The cleaving
process exposes flat BiO terraces at the surface, as the inter-unit-cell BiO-BiO bond is the
weakest one in the crystal structure. [48] Figure 3.4 shows a typical STM topograph and
26
conductance spectrum taken on this surface, both of which are comparable to data reported
in the literature for cleaved single crystal samples. [28, 29, 30, 31, 32, 49] Notably, the
topograph shows a clear atomic corrugation, and a b-axis superlattice distortion known to
run throughout every plane of the bulk. [50] The spectra show an inchoate energy gap and
pronounced peaks at about ±40mV , both characteristic of the density of states in the super-
conducting state of Bi2Sr2CaCu2O8+δ. However, cleaving thin films requires significantly
more mechanical force than cleaving single crystals, most likely because thin films contain a
higher density of defects. Whether the consequence of the thin film growth having more step
edges, and hence requiring more forceful cleaves, or of the more forceful cleaves resulting in
stepped surfaces themselves, the samples were found to BiO terraces 100− 1000A in extent
containing extended defect structures such as step edges and twin boundaries.
Cleaved thin film samples often contain BiO terraces terminated by step edge boundaries,
such as the one shown in the STM topograph in Figure 3.5a. Because tunneling conductance
spectra taken far away from this step edge (Fig. 3.6a) resemble the spectra taken on cleaved
single crystal samples, which are terminated by BiO surfaces, we can identify the terrace
on the top side of the step edge as being the BiO plane. From comparing the measured
distance between the top and bottom terrace to the known separation of the crystallographic
planes [50], the bottom terrace is shown to be the lower of the two CuO2 planes of the
Bi2Sr2CaCu2O8+δ crystal structure (Fig. 3.5b), which we will focus on in Chapter 5. This
means that the step edge terminates the upper of the two CuO2 planes of the unit cell.
To determine the angle that the step edge makes relative to the pair potential, a careful
registry of the atomic corrugations in the topograph are made. These corrugations most
likely represent Bi atoms in the surface BiO plane, and the Cu atoms of the CuO2 plane sit
directly beneath them. The step edge is seen to run at roughly a π/4 angle to the nearest
neighbor direction of the corrugations, which corresponds to a π/4 angle relative to the
Cu−O bond direction. Angle resolved photoemission spectroscopy has established that this
27
0
1
2
3
4
-100 0 100
0Å
8Å
16Å
24Å
32Å
40Å
48Å
56Å
Co
nd
ucta
nce (
pS
)
BA
Figure 3.4: (A) STM topographs of the BiO surface of a cleavedBi2Sr2CaCu2O8+δ thin film far from any defects resemble topographs onpristine single crystals (100A × 100A area imaged with I = 200pA andV = −200mV ). (B) Spatially-resolved conductance spectra taken along theline indicated in (A) resemble published data taken on single crystals (dc junc-tion impedence Rj = |−200mV |/200pA = 1GΩ, the data are offset for clarity).[28, 29, 30, 31, 32, 49] Both figures have been adapted from Ref. [34].
28
+
-
(0,π
)
(π,π)
11
0
ste
p e
dg
e
Normal Superconducting
02468
0 25 50 75 100
0Å
12
Å2
5Å
37
Å5
0Å
75
Å
A B C
Lateral Distance (Å)
He
igh
t (Å
)
(0,π
)
(π,π)
BiO
SrOCuO
Ca
Figure 3.5: (A) This STM topograph shows a step edge oriented at a π/4 tothe underlying lattice (I = 50pA and V = −200mV ). It has been renderedin a way to bring out the atomic corrugations on both the high (right) andlow (left) sides of the step edge. (B) This graph shows a single line scanfrom the topograph in (A) overlayed with the known separation between thecrystallographic planes of the Bi2Sr2CaCu2O8+δ. The spectra in Figure 3.6awere taken at the positions indicated by the arrows in (B) and the boxes in(A). (C) A schematic diagram of the subsurface CuO2 plane can be createdfrom the atomic positions in the topograph. All figures have been adaptedfrom Ref. [34].
corresponds to the nodal direction of the pair potential. [51] We thus have a situation similar
to the schematic diagram in Figure 3.5c, where a 110 step edge terminates the edge of a
CuO2 plane.
The interruption of the CuO2 lattice at the step edge should serve as a strong pertur-
bation to the superconducting states which exist on this plane. First, this should suppress
the superconducting order parameter strongly within a coherence length of the step edge.
[14, 15] Intuitively, the spectra should thus resemble the V-shaped spectra typical of conduc-
tance spectra taken over pristine samples at temperatures well above the superconducting
transition. [49] Although this spectrum has been interpreted to result from either inelastic
scattering [52] or a non-Fermi liquid ’normal’ state [53], superconducting order is unambigu-
ously suppressed in these spectra. What’s more, the suppression of the order parameter
29
0
1
2
3
50Å
37Å
25Å
12Å
0Å
-100 0 1000
1
2
3
4
5
6
12Å
25Å
37Å
50Å
62Å
0Å
-100 0 100
Energy (mV)
A B
Co
nd
ucta
nce (
pS
)
Figure 3.6: Spatially resolved STM spectra taken along a line perpendicularto (A) and along (B) the 110 step edge in Figure 3.5 are shown. For both(A) and (B), the dc junction impedence was Rj = | − 200mV |/200pA = 1GΩ,and the lines indicate the zero conductance level for the spectra, which havebeen offset for clarity. The figures have been adapted from Ref. [34].
near the interface raises the possibility that quasiparticles in this region can undergo An-
dreev reflection from the pair potential of the superconductor. [3] For a step edge oriented
at a π/4 angle with respect to the pair potential, quasiparticles reflected by the step edge
will be retro-reflected by a pair potential of opposite sign for successive Andreev reflections.
Moreover, for this orientation, there should be constructive interference between successively
Andreev reflected quasiparticles for all possible quasiparticle trajectories. We therefore ex-
pect scattering from the step edge in Figure 3.5 to give rise not only to a suppression of the
order parameter, but also to the appearance of an Andreev bound state.
We used STM conductance spectra to examine the effects of quasiparticle scattering on
30
the local density of states near the step edge. Spectra taken perpendicular to the step edge
(Fig 3.6a) clearly demonstrate both the consequences arising from quasiparticle scattering at
the step edge we had anticipated. Starting about 30A from the step edge, the conductance
peaks at about ±40mV begin to decrease in amplitude, and, concomittently, the in-gap con-
ductance increases. These spectra as a function of distance are remarkably similar to spectra
taken over pristine samples at temperatures well above the superconducting transition. [49]
The similarity of these two sets of spectra indicate that superconducting order is completely
suppressed near the step edge. In addition, within a coherence length of the step edge,
a well-defined zero bias peak appears in the tunneling conductance. Conductance spectra
taken alongside the step edge (Fig. 3.6b) confirm that this state persists for several coher-
ence lengths next to the step edge. This observation confirms that a novel one-dimensional
zero-bias state is bound to step edges running along the nodal directions of the pair poten-
tial. The spatial mapping of the suppression of superconducting order and the formation of
the Andreev bound state near an extended 110 defect are the central new results reported
in this section.
The results reported here on the density of states near 110 step edges bear a strong
resemblance to tunneling results on 110 surfaces. [3] Although the geometry for quasi-
particle scattering is similar in both sets of experiments, the geometry of the experiments
themselves is different. Here, we are tunneling into the states at a 110 interface in the
out-of-plane direction, whereas previous experiments have tunneled in-plane into a 110
oriented surface. Our results confirm that the Andreev scattering modifies the states at
the interfaces, rather than only influencing tunneling into the sample on in-plane directions.
Our results also add considerable detail to earlier reports of the Andreev bound state’s sen-
sitivity to disorder. [26, 27] In previous experiments on planar tunnel junctions, the zero
bias conductance peak was shown to be suppressed in junctions where the 110 surface
had been disordered from ion bombardment. [26] Here, the sensitivity to disorder manifests
31
itself in the spatial inhomogeneity of the peak. As seen in Figure 3.6b, although the peak
extends for many coherence lengths along the step edge, it vanishes at other locations. The
data does not, however, show a clear correlation between disorder in the topography and the
suppression of the zero bias state.
Because it can find Andreev bound states at strongly scattering defects, STM can be
used to characterize extended defects about which little is known. Consider the 110 twin
boundary shown schematically in Figure 3.7a. This interface has the same qualities that led
to the formation of the Andreev bound state at the 110 step edge, provided, as has been
suggested [35, 36], that it strongly scatters quasiparticles. In fact, this geometry is similar
to aligned grain boundary junctions, where a zero-bias conductance peak has been seen. [18]
In Figure 3.7b, we show an STM topograph of a twin boundary that has this geometry.
However, spectra taken near this twin boundary (Fig. 3.7c) show no evidence either for the
suppression of superconducting order, nor of a zero bias conductance state. This naively
suggests that twin boundaries do not have a strong defect potential. An alternative view of
the twin boundary is that it is the visible symptom of a gradual atomic-scale strain field.
[54] Self-consistent calculations of the order parameter near a strain-induced twin boundary
show that d-wave superconducting order is not strongly suppressed nearby. [37] This view
is supported by the conductance spectra taken near the twin boundary in Figure 3.7.
Finally, previous theoretical works have raised the question whether secondary super-
conducting order parameters can stabilize at extended scattering interfaces. [55, 35, 36, 56]
Previous experiments on 110 surfaces have observed a splitting of the zero bias conduc-
tance peak below T ≈ 7K under zero applied field, which was interpreted to result from
the formation of a subdominant complex order parameter at the interface. [16, 22] Within
our experimental resolution (±3mV ), we have not seen a splitting of any of the zero bias
conductance peaks we have found near step edges at T = 4.2K. The width of the zero-
bias conductance peaks in this work (±15mV ) exceeds both the width of the unsplit peaks
32
B
-100 0 100
1
2
3
0Å
25Å
50Å
Energy (mV)
Co
nd
uc
tan
ce
(p
S)
C
+
-
110 twin boundary
Normal
Superconducting
b
a(0,π)
a
b(0,π)
A
Superconducting
Figure 3.7: (A) This is a schematic representation of a 110 twin boundary.A sample quasiparticle trajectory, and the pair potential, is also shown. (B)This is an STM topograph of a twin boundary similar to the one sketchedschematically in (A). It is identifiable as a 110 twin because of the per-pendicular b-axis supermodulation on both sides of the twin (I = 200pA andV = −200mV ). (C) These STM spectra were taken along the twin boundaryevery 25A (Rj = | − 200mV |/200pA = 1GΩ). There is some evidence for theformation of an in-gap kink in the conductance at some (denoted by arrows),but not all (the middle trace) locations near the twin boundary. The zero con-ductance level of the spectra, which have been offset for clarity, is indicatedby the bar. These figures have been adapted from Ref. [34].
33
(±5mV ) and the splitting of the peak (±2mV ) seen in the planar tunnel junction experi-
ments, and may thus be masking the splitting. The width of the zero bias conductance peak
has been proposed to be related to the roughness of the interface. [27] However, calculations
show that a roughness much larger than we observe would be required to explain the mea-
sured width. We believe a more likely explanation is that the width of the peak seen here
arises from a reduced lifetime for the Andreev bound state, caused by scattering from the
atomic-scale disorder at the step edge.
The possibility of nucleating a sub-dominant order parameter at 110 twin boundaries
has also been raised, although most models have assumed a strong suppression of supercon-
ducting order near the twin boundary. [35, 36, 56] However, calculations treating the twin
boundary as an atomic-scale strain field have also shown that a real subdominant order pa-
rameter with s-wave symmetry mixes with the dominant d-wave order parameter near twin
boundaries. [37] Although one would expect that the admixture of a subdominant s-wave
order parameter, which introduces a finite amplitude energy gap along the nodal direction of
the d-wave gap, would produce a hard energy gap in the density of states, calculations have
shown the low-energy density of states to remain unperturbed in this situation. [37] However,
they find the subdominant order parameter producing in-gap kinks in the density of states.
[37] Consistent with the calculations, the STM conductance spectra show no evidence for a
hard energy gap at low energies, and do show evidence for in-gap kinks at some positions
near the twin boundary. More definitive spatial mapping of the tunneling conductance will
be required to determine why the in-gap kinks disappear at other locations along the twin
boundary, and, ultimately, whether stronger evidence can be found for this sub-dominant
order parameter.
34
0.1
0.2
0.3
-0.4 -0.2 0 0.2 0.4
dI/
dV
(n
S)
Energy (V)
A B
Figure 3.8: (A) STM topographs of the BiO surface of a cleavedBi2Sr2CaCu2O8+δ single crystal taken at 40K (380A × 380A area imagedwith I = 20pA and V = −200mV ) resemble those taken at liquid heliumtemperatures. (B) Typical conductance spectra also show the same features-sharp peaks at ≈ ±40mV and an inchoate energy gap- as those taken at liquidhelium temperatures (dc junction impedence Rj = | − 150mV |/40pA).
3.3 Local DOS Modulations
In the previous section, we have examined how the density of states changes on the atomic
length scale near spatially extended defects. We found that a novel zero-energy state forms
as a consequence of the phase of the d-wave pair potential near extended defects that strongly
scatter quasiparticles. Here, we will examine the spatially-resolved density of states far away
from these defects. Surprisingly, we find that the low energy states form dispersing periodic
patterns in real space. [38, 39, 40, 41] This dispersion is found to be the consequence of weak
scattering, which is sensitive to the amplitude of the pair potential. [42, 43, 44, 45, 46, 47]
We have examined the spatial structure of the density of states far away from extended de-
fects in the superconducting state. [38] Slightly underdoped single-crystal Bi2Sr2CaCu2O8+δ
samples (TC = 85K) were mechanically cleaved along the c-axis direction in ultra-high vac-
uum at room-temperature prior to performing measurements with a home-built variable-
temperature STM (VTSTM) at T = 40K. Single crystals were used for this study because
cleaved samples rarely contain extended defects, and flat, undisturbed terraces can extend
35
for > 10000A. Atomic-resolution topographs of the cleaved surface show the atomic corruga-
tions and the b-axis superlattice distortion familiar from low-temperature measurements (Fig.
3.8a). Energy resolved conductance spectra taken at a fixed position on the sample reveal a
density of states typical of that measured in the superconducting state of Bi2Sr2CaCu2O8+δ
(Fig. 3.8b). We will focus on the tunneling conductance measured at a fixed energy as a
function of position, which maps the spatial arrangement of the electronic states at a given
energy. The local density of states (LDOS) maps measured for energies inside the energy
gap, shown in Figure 3.9, contains periodic structures, including an unexpected modulation
running at 45 to the b-axis supermodulation.
To analyze periodicity of the modulated structures seen in the LDOS maps, a two-
dimensional Fourier analysis was performed. Limits on the tunneling conductance were
imposed to limit the effects of the two low-energy resonances in the data in the FFTs
(25 − 55pS for 7mV , 50 − 175pS for 13mV , 75 − 175pS for 20mV , and 90 − 190pS for
13mV . The power spectrum (Fig. 3.9b) contains three main features. The broad feature
near the center corresponds to long-wavelength changes in the density of states associated
with random dopant fluctuations. [57] The two sharp peaks flanking the center peak at
a distance of 2π/6.8a0 along the 〈π, π〉 direction are associated with the structural b-axis
supermodulation. Finally, the two broad peaks along the 〈0, π〉 (the CuO bond direction)
correspond to the unexpected incommensurate modulation we will focus on. Analysis of
maps acquired simultaneously at different in-gap energies shows that the wavevector of the
modulated patterns changes as a function of energy (Fig. 3.10a). Similar modulations have
been seen by other STM studies performed at low temperatures (T = 4K) for a wide range
of dopings (Fig. 3.10b&c). [39, 40] The low temperature experiments see a feature along
the 〈0, π〉 direction that disperses in a fashion similar to the dispersion we have measured at
T = 40K. (Fig. 3.10a) However, these T = 4K studies also see a dispersive feature along
the 〈π, π〉 direction, and a high-resolution data set [40] has identified a total of 5 dispersive
36
A B
42mV 42mV
q
s
(0,0) <π,π>
<0,π>
Figure 3.9: (A) Spatially-resolved conductance maps (Rj = |−150mV |/75pA),were acquired simultaneously at four energies between 7− 23mV on the same380A × 380A region as Fig. 3.8a. The greyscale extends from 200pS (white)to 25pS (black). The 42mV conductance map, which contains modulationsarising only along the b-axis supermodulation, was taken on a different areameasuring 550A× 550A (Rj = | − 150mV |/40pA, greyscale map extends from36pS to 136pS). (B) FFTs were performed on the data in (A) to highlight thedifferent wavevectors contributing to the modulated patterns. The greyscaleextends from 1.88pS (black) to 0pS. Highlighted are the contribution fromthe b-axis supermodulation (S) and the contribution oriented at 45 to thisdirection (Q).
37
3
4
5
6
0 20 40Energy (mV)
Wav
ele
ng
th (
a0)
A B C
7
1
45
3
6
Figure 3.10: (A) The Fourier analysis performed on the data in Figure 3.8 wasused to determine the energy-dependence of the wavevector Q running alongthe 〈0, π〉 direction. Plotted here is the wavelength of the patterns measuredby us at T = 40K (black squares) and the dispersion measured by Ref. [39] atT = 4K (grey). The wavelength is in units of a0 = 3.8A, the nearest neighborCu−Cu distance. (B) The data at T = 4K, shown here at V = 16mV , showtwo sets of strong peaks- along the 〈0, π〉 (blue) and 〈π, π〉 (red) directions-and four weaker peaks (black). (C) All the peaks seen at T = 4K were foundto disperse. Parts (B) and (C) of this figure were adapted from Ref. [40].
features in the power spectrum of the LDOS not seen at T = 40K (Fig. 3.10d). The features
seen at T = 4K have been proposed to result from a simple elastic scattering interference
model. [43, 39, 40] We will now describe this model, and demonstrate how it applies to our
data at T = 40K.
Electron waves elastically scattering off defects lead to periodic modulations in the density
of states. A well-studied example of this is the modulated pattern found near step edges
and point defects on certain noble metal surfaces, as discussed in Chapter 2. [58, 59, 60,
61, 62] Recall that the set of wavevectors ~q where there is some correction to the density of
states arising from elastic scattering interference can be determined very simply using toy
models. The easiest one to illustrate for the case of Bi2Sr2CaCu2O8+δ is the particle-based
toy model. Quasiparticles exist on curves of constant electron energy in ~k-space, which,
in the superconducting state of Bi2Sr2CaCu2O8+δ, are a set of “banana”-shaped curves
38
1
5
7
3
4
6 (0,π)
(π,π)
Figure 3.11: This is a model representation of the curves of constant elec-tron energy in ~k-space in the superconducting state of Bi2Sr2CaCu2O8+δ atω = ∆/3 (red) and ω = 2∆/3 (blue). Also shown are the 6 wavevectors ~qthat connect points on these curves where Ref. [40] sees peaks in the Fouriertransform of the STM density of states.
centered around the 〈π, π〉 direction in the first Brillouin zone (Fig. 3.11a). To determine
the wavevectors ~q where we expect modulations arising from elastic scattering, we find the set
of wavevectors ~q which connect points on these curves in ~k-space. We would naively expect
that the wavevectors ~q joining the points on these curves with the highest joint density of
states would have the largest contribution to elastic scattering. Since the total density of
states at a given energy E is given by∫
ε(~k)=Ed~k
|5ε(~k)|, we expect the points on the curves where
the integrand is largest to contribute the most. Thus the largest contribution from elastic
scattering should arise at the wavevectors connecting the ends of the “bananas”. [39] Indeed,
the 6 unique vectors predicted by this method are the same ones where high-resolution STM
measurements at 4K have seen density of states modulations. [40] However, the toy model
cannot simply describe why only one of these 6 remains in our measurements at 40K.
To more accurately determine the density of states correction arising from elastic scatter-
ing interference, we have adopted the perturbative treatment of Ref. [63]. A more complete
explanation appears in Appendix B. In general, many-body problems in condensed matter
39
physics are often solved by finding the Green function that solves a Hamiltonian equation
of motion. We take the total Green function G(~r, ω) near an impurity to be the sum of
the Green function over pristine regions G0(~r, ω), plus some correction due to scattering
off the impurity. To leading order, the correction to the Green function arising from elas-
tic scattering takes the form∫
d2~r1G0(~r − ~r1, ω)V (~r1, ω)G0(~r1 − ~r, ω). This is the limit of
weak scattering, or Born Scattering (Fig. 3.12). The density of states in a superconduc-
tor is n(~r, ω) = − 1πImG(~r, ω) − F (~r, ω), where G is the single-particle Green function
and F is the anamolous Green function that contains the coherence factors responsible for
superconductivity. Elastic scattering interference thus introduces a correction
δn(~r, ω) = − 1
πIm
∫
d2~r1G0(~r − ~r1, ω)V (~r1, ω)G0(~r1 − ~r, ω)
±F0(~r − ~r1, ω)V (~r1, ω)F0(~r1 − ~r, ω) (3.1)
to the density of states, where G0 is the single particle Green function, F0 is the anamolous
Green function (F0 = 0 in the absence of particle-hole correlations), V is a weak, finite-
range scattering potential, and the minus (plus) sign corresponds to potential (magnetic)
scattering. [46, 43, 44, 64, 45, 65] The Fourier transform of the Born correction δn(~q, ω) =
− 1πImV (~q, ω)Λ(~q, ω) separates into a part
Λ(~q, ω) =∫
d2~kG0(~k, ω)G0(~k + ~q, ω) ± F0(~k, ω)F0(~k + ~q, ω) (3.2)
that contains all the wave interference information, and a part
V (~q, ω) =∫
d2~xe−i~q·~xV (~x, ω) (3.3)
that acts like a static structure factor. [44] Assuming that the scattering potential is a single
point impurity in real-space, the structure factor acts like an all-pass filter, and the density
of states correction depends only on the wave interference term Λ. Following previous works,
[43, 44, 64, 45] we model the superconducting state with the Green functions,
G0(~k, ω) =ω + iδ + ε~k
(ω + iδ − ε~k)(ω + iδ + ε~k) − ∆2~k
40
A B C
Figure 3.12: (A) This is the diagram corresponding to (weak) Born scattering.(B and C) These types of diagrams have been neglected in the treatmentpresented.
F0(~k, ω) =∆~k
(ω + iδ − ε~k)(ω + iδ + ε~k) − ∆2~k
(3.4)
where ε~k = 120.5mV − 595.1mV × (cos kx + cos ky)/2 + 163.6mV × cos kx cos ky − 51.9mV ×
(cos 2kx+cos 2ky)/2−111.7mV ×(cos 2kx cos ky +cos kx cos 2ky)/2+51mV ×cos 2kx cos 2ky is
the Bi2Sr2CaCu2O8+x band structure from ARPES [66] for slightly underdoped (x = .12)
samples, ∆~k = 45.0 × (cos kx − cos ky)/2 is the superconducting gap function, and δ is a
broadening term.
The density of states corrections arising from scattering interference provide a promising
explanation for the modulations seen in the STM data at both T = 4K and T = 40K. As
shown in Figures 3.13a&b, the formalism yields density of states corrections with sharp peaks
in ~q-space along the 〈π, π〉 (〈0, π〉) direction for potential (magnetic) scattering. The presence
of the sharp peaks is thus understood to be a consequence of the anamolous Green function,
arising from the coherence factors for the superconducting state. [46] By comparison, the
STM studies performed at T = 4K [39, 40] contain peaks along both the 〈π, π〉 and 〈0, π〉
directions (Fig. 3.13c), implying that a combination of magnetic and potential scattering
are required to reproduce the observed data. In total, the calculations observe all 6 features
identified in the high resolution data taken by Ref. [40]. At T = 40K, however, the STM data
(Fig. 3.13f) [38] shows peaks only along the 〈0, π〉 directions. Scattering interference patterns
calculated with δ = 7mV , appropriate for the experimental conditions under which the data
41
q=(0,0) (0,π)
(π,π)
A B DC
Figure 3.13: We calculated the power spectrum of δn(~q, ω) in the first Brillouinzone for the superconducting state. Dark regions correspond to a larger densityof states correction than lighter regions. The power spectrum of δn(~q, ω =16mV ) in the first Brillouin zone shows big peaks (denoted by black arrows)along the 〈π, π〉 directions for pure potential scattering (A), and along the〈0, π〉 directions for magnetic scattering (B). In addition, the four additionalweaker peaks seen in the STM data at 4K are identified here by the blackcircles. These patterns were calculated using a broadening of δ = 1mV . Thecalculation was repeated for an energy broadening of δ = 7mV , and is shownhere in (C) for pure potential scattering and in (D) for magnetic scattering.The only feature which remains strong is the 〈0, π〉 feature, denoted by theblack arrow.
was measured, demonstrate that the other peaks have washed out at this level of energy
broadening, while the 〈0, π〉 mode remains (Fig. 3.13d). The reason is relatively simple- two
of the modes, one along the 〈0, π〉 direction and the other along the 〈π, π〉 direction, have
a significantly larger amplitude than the others. The 〈π, π〉 mode gets washed out because
it disperses faster than the 〈0, π〉 mode, meaning that a smaller level of energy broadening
leads to a more significant broadening in ~q-space for the former in comparison to the latter.
The most distinctive characteristic of the peaks in the STM data at T = 4K and T =
40K, and the calculations in the superconducting state, is that their wavelength changes as
a function of energy. Because the peak along the 〈0, π〉 direction appears at both T = 4K
and T = 40K, we focus on its dispersion here. As can be seen in Figure 3.14a, the calculated
dispersion agrees with the measured dispersion for higher energies (|ω| > 15mV ). The
dispersion can be understood on qualitative grounds by returning to the particle-based toy
model discussed earlier (Fig. 3.14b). Quasiparticles of a given energy exist on banana-
42
shaped curves of constant electron energy in the superconducting state. The characteristic
wavevector for scattering interference are the ones joining the tips of these “bananas”. At
different energies, the shape of these “bananas” changes predominantly because of the pair
potential, as changes in the band structure tend to be gradual over small energy ranges.
As a result, the wavevector for scattering interference can be used to map the amplitude
of the pair potential. Indeed, the dispersion of the peaks seen in the data at T = 4K has
been shown previously to be consistent with the amplitude of the d-wave pair potential (Fig.
3.14c). [39] However, the disagreement between the data and the scattering interference
model at low energies (|ω| < 15mV ) is significant. The size of the scattering wavevector at
the Fermi energy is essentially independent of the details of the scattering, as the available
phase space for scattering is restricted to the nodal points of the Fermi surface, where the
energy gap has zero amplitude. This is exactly where the data and the scattering interference
model disagree on the size of the scattering wavevector the most. It has been suggested in
the literature [41, 67, 68] that the disagreement stems from the fact that the modulated
patterns no longer disperse at low energies, and that this lack of dispersion represents some
form of ordering. In this scenario, the dispersion seen at higher energies, or along different
directions, is due to scattering off this order. Although more work is needed to interpret the
almost non-dispersing low-energy data, it now appears clear that the dispersion seen in the
superconducting state data arises from elastic scattering interference.
Here, we have presented a case for quasiparticle scattering being consistent with the
periodic modulations seen in real-space maps of the density of states by STM in the super-
conducting state. The energy dependence of the wavelength of these modulations appears
to be sensitive to the amplitude of the d-wave pair potential for the HTS. Although the
data taken by us at T = 40K contains defects, as evidenced by the low-energy resonances
in the data of Figure 3.9, the modulated patterns in the density of states do not appear to
be pinned by these defects. Moreover, data taken by other groups at T = 4K show these
43
B
2
3
4
5
6
-40 -20 0 20 40Energy (mV)
Wav
ele
ng
th (
a0)
A C
k=(0,0) (0,π)
(π,π)
q
Figure 3.14: (A) The dispersion for the calculated 〈0, π〉 mode (solid line)compares favorably with the dispersion measured for this feature at T = 40K(squares, from Ref. [38]) and T = 4K (triangles, diamonds and circles, fromRef. [39]) for energies ω > 15mV . (B) Here we show the curves of constantelectron energy for Bi2Sr2CaCu2O8+δ in the superconducting state at ω =∆/3 (blue) and ω = 2∆/3 (red). The characteristic wavevector for scatteringat the two energies along the 〈0, π〉 direction is also shown. (C) This dispersionhas been used by Ref. [40] to reconstruct the angular dependence of the energygap, and is shown to be consistent with a d-wave form (Figure from Ref. [40]).
modulated patterns, but show no obvious scattering site. We suggest this might result from
the scattering potential, which causes these modulations, being either very weak or extended
in space. [69] True bound states, which would produce a discernable resonance in the LDOS
maps, are expected only in the unitary, or strong, scattering limit. [63] Conversely, weak
scattering might produce modulated LDOS patterns without producing any in-gap resonance
feature.
3.4 Applications
Although the experiments in this chapter do not lead to a new understanding of the su-
perconducting state of the HTS, they present us with a set of tools that will be useful in
examining parts of the phase diagram where the electronic state is not well-understood. In
particular, examining the density of states close to scattering sites has been demonstrated
to sensitively probe the electronic state of the system.
44
The ability to detect Andreev bound states locally has practical value as a phase-sensitive
test for d-wave, BCS-like superconducting quasiparticles. [25] Here we have shown that An-
dreev bound states should be bound to 110 step edges on a sample that was known to have
d-wave superconducting order. In many situations throughout the cuprate phase diagram,
it is unclear whether local, fluctuating superconducting correlations are present, even when
global superconducting order is not. For example, local, fluctuating superconductivity has
been postulated as an explanation for the so-called pseudogap in the density of states for
underdoped samples at T > TC , the subject of the next chapter. In addition, the electronic
phase of the subject of the last chapter of this thesis, a bare CuO2 plane at the surface of a
HTS, is undeterminable from a simple examination of the density of states. In both cases,
a careful search for a zero bias conductance peak in the density of states near a 110 step
edge could be used to establish whether d-wave superconductivity is present.
The scattering interference scenario can be used to characterize any modulated pattern
seen in the density of states by STM. Here, we have shown that modulated patterns in
the LDOS measured in a superconducting sample disperse. This dispersion arises from
weak scattering interference, which probes the amplitude of the d-wave pair potential in the
superconducting state. In many situations, such as the pseudogap regime of the HTS, the
electronic state of the system is not known. Comparison of the modulated patterns measured
in STM data with calculations using the candidate Green functions for an electronic state
can help determine which of the candidates is consistent with the data. [46] Candidate Green
functions often differ only by correlations that subtly affect the density of states, but produce
completely different scattering interference patterns. In the next chapter of this thesis, we will
discover that STM experiments see LDOS modulations in the enigmatic pseudogap regime
of the HTS, and will attempt to compare the Green functions of various proposed electronic
states with the measured patterns using the scattering interference formalism developed in
the superconducting state.
45
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46
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49
Chapter 4
Local Ordering in the Pseudogap
Regime
4.1 Introduction
No other issue in the physics of the HTS exposes our lack of understanding of the microscopic
mechanisms underlying the entire phase diagram more than the so-called pseudogap regime
(Fig. 4.1a). This part of the phase diagram gets its name from a depression in the density
of states at the Fermi energy, which slowly fills in upon increasing temperature through an
ill-defined T ∗ for underdoped samples (Fig. 4.1b). Because this depression in the density
of states is symmetric about the Fermi energy, it intuitively appears to result from some
correlation effect. Whether this correlation originates from residual antiferromagnetism,
remnant superconductivity, or some other source remains a crucial open question. The
answer not only has bearing on the origin of the pseudogap, but also has the potential to
alter our view of the doped insulator, the superconducting state, and the whole cuprate
phase diagram itself.
The pseudogap in the density of states in the underdoped HTS above TC strongly in-
50
A B
Figure 4.1: (A) The doping-temperature phase diagram of the cuprates con-tains a superconducting dome centered at a doping of x = 0.16, where TC
reaches its maximum value (≈ 95K in Bi2Sr2CaCu2O8+δ). A symmetric de-pletion of electronic states at the Fermi energy called the pseudogap (B) isseen for underdoped samples (x ≤ 0.16) above TC . As temperature is raisedtowards T ∗, the pseudogap in the density of states diminishes, as states fill inthe gap. Although it remains unclear where exactly the T ∗ line lies, both thepseudogap and T ∗ increase monotonically with underdoping (from Ref. [1]).The only feature in the density of states which disappears at TC are the sharppeaks flanking the energy gap.
fluences any measurement that probes quasiparticle degrees of freedom. The most direct
measurement of gapped electronic states is perhaps provided by angle-resolved photoemis-
sion spectroscopy (ARPES), which measures the occupied part of the density of states as
a function of energy and momentum. For optimally doped samples at low temperatures,
ARPES sees that the density of states is gapped away from the Fermi energy with a d-wave
angular dependence. [2] For overdoped samples, the size of the energy gap scales with TC at
low temperatures, and the energy gap disappears above TC . This behavior is consistent with
the view that it is a manifestation of the amplitude of the pair potential, and that increasing
temperature destroys superconductivity by breaking Cooper pairs apart. Contrary to ex-
pectation, for underdoped samples, the energy gap continues to grow while TC is shrinking
51
(Fig. 4.2a). [3, 4] Even more mysteriously, as the temperature is raised through TC , ARPES
finds that the energy gap near the (0, π) points persists- the so-called pseudogap (Fig. 4.2b).
[5, 6] The size of the pseudogap decreases linearly with increasing temperture or hole con-
centration. Meanwhile, the locus of momenta with states at the Fermi energy widens from a
single point in the superconducting state, into a so-called Fermi arc in the pseudogap regime
((Fig. 4.2c)). [7, 8] As temperature is further raised, this arc widens to meet the zone edge,
and the pseudogap is destroyed. Upon including the observation that the quasiparticle states
in the pseudogap regime become ill-defined in energy and momentum [9], the ARPES data
appears to be a key ingredient in explaining the properties of underdoped HTS above TC , as
it can be phenomenologically tied to their thermal, electronic [10], and optical [11, 12, 13]
properties.
The underlying cause of the pseudogap in the density of states, however, remains elusive.
Complicating matters is the fact that the electronic state of the HTS in the pseudogap regime
may not even contain quasiparticles. The well-defined features measured by photoemission
as a function of energy and momentum in the superconducting state have been replaced
with very broad features in the pseudogap regime (Fig. 4.3). The ill-defined nature of the
electronic states in the pseudogap regime suggests that a fundamentally different picture
from that used to describe conventional metals is required.
The basic ideas underlying the current debate actually predate the discovery of the pseu-
dogap in the density of states. The first experiment to detect an abnormal signal in under-
doped samples above TC was nuclear magnetic resonance (NMR). [15, 16] In these exper-
iments, the nuclear spins of a particular species were first perturbed by a magnetic pulse,
and then the relaxation rates of the excited spins were measured. The relaxation rate of
Cu spins in overdoped samples drops from finite values above the superconducting tran-
sition, to near zero in the superconducting state. For underdoped samples, on the other
hand, there is a gradual decrease of the relaxation rate starting at temperatures well above
52
A B
C
Figure 4.2: (A) In the superconducting state, ARPES measures an energy gapwhose maximum value increases while TC decreases for underdoped samples(from Ref. [3]). (B) As underdoped samples are warmed up through TC , theenergy gap persists to higher temperatures. Shown is the density of statesalong the gap maximum (〈0, π〉) direction as a function of energy (from Ref.[6]). (C) This is a schematic diagram of the Fermi surface measured at differenttemperatures in an underdoped sample. Here, Γ − M refers to the 〈0, π〉direction and Γ−X and Γ− Y refer to the 〈π, π〉 directions. The left panel isfor T < TC , the middle panel is between TC and T ∗, and the right panel is forT > T ∗ (from Ref. [7]).
53
A
A
B
B
Figure 4.3: Shown is the photoemission intensity as a function of electronenergy on the Fermi surface along the 〈π, π〉 direction (A) and the 〈0, π〉 direc-tion (B). In the superconducting state of a slightly underdoped sample (top),angle-resolved photoemission measures sharp quasiparticle features. In con-trast, above TC (bottom), the peak-like features have become broad enoughthat their position is now comparable to their width. (Figures adapted fromRef. [14]).
54
the superconducting transition. The discovery of this effect led to the claim that the effect
was due to some form of precursor superconductivity [15]. Almost immediately afterwards,
another group claimed the Cu relaxation rates are enhanced by local fluctuating spin order
[16]. Either phenomenon could produce a pseudogap in the density of states. The pseudo-
gap could be a signature of Cooper pairing occuring above TC , with the pairs not becoming
phase coherent until the temperature is decreased below TC . Pseudogap-like depressions in
the density of states can also result from various forms of ordering. Since the initial NMR
experiments, compelling experimental evidence has suggested that both these phenomena
are present above TC for underdoped samples. Simultaneously, various theoretical proposals
have been raised to explain the abnormal behavior of the underdoped HTS above TC .
4.1.1 Precursor Superconductivity Experiments
Although neither zero resistivity nor the Meissner effect have ever been seen in HTS samples
in the pseudogap state, other probes indicate that some fluctuating form of superconductivity
might be present in the pseudogap regime. In Nernst effect measurements, a magnetic field
B is applied along the z direction, a thermal gradient −5 T along the x direction, and the
electric field E develops along the y direction is measured as a function of temperature and
doping (Fig. 4.4a). [17] Normal metals have a very small Nernst voltage, originating from the
motion of charged carriers. The thermal gradient causes a diffusion of electrons from the hot
end of the sample to the cold end. Because there can be no net current flow, this sets up an
electric field that causes a perfectly matching counterflow of electrons in the other direction.
Application of a vertical magnetic field causes a portion of these currents to deflect sideways.
These sideways currents almost cancel, leading to a small voltage being established to ensure
no net current flow sideways. In contrast, in type-II superconducting samples, like the HTS,
an applied magnetic field generates vortices, regions of normal material that thread magnetic
flux and carry no charge. If the vortices are mobile, the thermal gradient makes them move
55
from the hot end of the sample to the cold end. The motion of these vortices generates a
relatively large electric field perpendicular to both the direction of vortex motion and applied
magnetic field from the Josephson effect (Fig. 4.4b). The data indicates that the Nernst
signal is very large in the superconducting state, and drops to a value one hundreth of its
value in the superconducting state at high temperatures. However, its value does not drop
off steeply at TC for underdoped samples- it remains anomalously large into the pseudogap
regime (Fig. 4.4c). The authors interpret their results as meaning that vortices, and hence
fluctuating superconductivity, exist above TC for underdoped samples even though the entire
sample is not superconducting. Although the temperature above which the Nernst signal
drops below a certain level does not track T ∗ throughout the phase diagram [18], fluctuating
superconductivity appears to exist even outside the superconducting dome for underdoped
samples.
Optical conductivity measurements have been used to measure the time scale for this
fluctuating superconductivity. A fast probe might be required to see properties of a rapidly
fluctuating superconductor, which might be inaccessable to DC probes like resistivity or
magnetization. [19] The frequency dependent conductivity of a superconductor σ(ω) =
σQkBTΘ(T )/ω, where TΘ is a measure of the stiffness of the phase of the order parameter,
σQ = e2/d, and d is the distance between superconducting planes. In the pseudogap state,
optical conductivity should approach the DC conductivity at low frequencies, and if fluc-
tuating superconductivity is present, the superconducting form at high frequencies. The
frequency where one behavior crosses over into the other one, Ω, is a measure of the time
scale for superconducting fluctuations (Fig. 4.5a). Upon normalizing the frequency to Ω, the
normalized conductivity |σ| = σQkBTΘ(T )/Ω(T ) was seen to collapse to a universal curve for
all frequencies and at temperatures both above and below the superconducting transition.
From analyzing a set of conductivity curves taken as a function of temperature at different
frequencies, both the phase stiffness energy Θ and the lifetime of the superconducting fluc-
56
∇ΤΕ
Β
V
∇Τ
Β
V
A C
B
e
Figure 4.4: This is a schematic diagram (top view) of a Nernst effect experi-ment in a normal metal (A) and a superconductor (B). The transverse voltageis small for (A), but can be comparitively large for (B). (C) The temperaturedependence of the Nernst signal ν = Ey/Bz|5T | minus the background signalfrom the motion of charged carriers at elevated temperatures is shown here fora wide range of dopings. The arrows indicate the TC for the given sample. Forthe overdoped sample (x = 0.17), the Nernst sigal becomes undetectable atTC , while for underdoped samples (0.06 < x < 0.12) the Nernst signal persistswell above TC (Figures from Ref. [17]).
57
A B
Figure 4.5: (A) The optical conductivity of a HTS has a universal scalingbehavior both above and below the superconducting transition (TC = 74K forthis sample). (B) From this scaling analysis, the temperature dependent phasestiffness energy TΘ and the lifetime for superconducting fluctuations τ = 1/Ωcan be determined as a function of temperature. Both these quantities remainfinite for temperatures above TC = 74K, but vanish below the temperatureT ∗ ≈ 170K (Figures adapted from Ref. [19]).
tuations τ = 1/Ω can be extracted as a function of temperature (Fig. 4.5b). For underdoped
samples, the phase stiffness energy does not abruptly fall to zero above the superconduct-
ing transition, but rather persists to higher temperatures. Furthermore, the time scale for
these superconducting fluctuations is small, but detectable, to similarly elevated tempera-
tures. This seems to support the idea that superconductivity exists on short time scales well
into the pseudogap regime, although once again, the characteristic temperature where these
effects disappear was identified to be smaller than T ∗.
4.1.2 Spin Order Experiments
Neutron scattering experiments have been used to characterize the spin structure of the
HTS throughout the phase diagram, and see remnants of antiferromagnetism above TC for
underdoped samples. Neutron scattering measures the energy shift and wavevector change
of a neutron that scatters off the atomic scale spin structure of a sample, which encodes
the momentum and energy of spin excitations in the material. The elastic signal (no energy
58
BA C D
Figure 4.6: (A) This is a schematic diagram of the peaks in ~k-space measuredby elastic neutron scattering in the undoped insulator. The Bragg peaks arelabeled as squares, and the peaks resulting from antiferomagnetic alignmentare labeled as circles. (B) This is the real-space schematic diagram correspond-
ing to the ~k-space picture in (A). Here, each bright spot corresponds to a Cusite, with red sites being spin up, and blue sites being spin down. (C) This
is a ~k-space diagram of the inelastic spectrum at higher energies. (D) This is
the real-space picture corresponding to the ~k-space diagram in (C). Here, redsites correspond to up spins, and blue sites correspond to up spins.
transfer) encodes the spin configuration of the ground state. In the undoped antiferromag-
netic insulator, neutron scattering sees the Bragg peaks for the Cu sites, which have localized
spins on them. A second set of peaks, at ((2n + 1)π, (2m + 1)π), arises from the breaking
of the square lattice symmetry due to the antiferromagnetic alignment of these spins (Fig.
4.6a&b). [20, 21, 22] In the inelastic spectrum, the peaks centered at (π, π) split, with in-
creased splitting at higher energies (Fig. 4.6c&d). [23, 24] This corresponds to a real-space
modulation of the spins when the neutron transfers energy to the spin lattice- the excita-
tion of spin waves on the antiferromagnetic ground state. For underdoped samples, neutron
scattering sees remnants of these features above TC in both the elastic and inelastic channels.
Data from one family of the HTS compounds, the Lanthanum cuprates, shows evidence
for ordered spins in the ground state for underdoped samples above TC . In this part of
the phase diagram, elastic neutron scattering shows that the elastic peaks present in the
undoped antiferromagnetic insulator at (π, π) are split along the nearest neighbor direction
59
δ(r.l.u.)
A
2δ
B C
Figure 4.7: (A) This is a schematic diagram of the elastic neutron scatteringspectrum on the Lanthanum compounds. (B) This is the real-space picture
complementary to the ~k-space diagram in (A). Here, black sites correspondto down spins, white sites correspond to up spins, and the grey regions inbetween are spinless. (C) The splitting of the peaks centered at (π, π) scaleslinearly with doping for underdoped samples. This corresponds to a decreasingdistance between the antiferromagnetic domains in (B) as holes are doped intothe system (Figure adapted from [25]).
(Fig. 4.6a,b). [30, 28, 25] This corresponds to a modulation in real space of antiferromagnetic
domains, reminscent of a spin wave, in the ground state of these compounds. Moreover, the
splitting of the peaks increases linearly with doping for underdoped samples, suggesting
that the size of the antiferromagnetic domains shrinks as holes are added (Fig. 4.6c). [25]
However, although the amplitude of the elastic signal disappears for overdoped samples above
TC [31], there is no correlation between the elastic signal measured by neutron scattering
and the pseudogap temperature scale T ∗. [32] More importantly, no elastic signal has yet
been detected in any of the non-Lanthanum HTS at superconducting dopings.
Inelastic neutron scattering data from all the HTS compounds shows evidence for residual
antiferromagnetic correlations above TC for underdoped samples. In the undoped insulator,
the elastic peaks at ((2n+1)π, (2m+1)π), characteristic of having an antiferromagnetically
aligned lattice, split at higher energies. [23, 24] Upon increasing doping to superconducting
dopings, the peaks at ((2n + 1)π, (2m + 1)π) move to a higher energy Er that scales with
TC . [33, 34] In the superconducting state, at energies both above and below Er, the peak
splits in a fashion similar to the spin wave in the undoped antiferromagnet (Fig. 4.8a).
60
δ(r.l.u.)
δ(r.l.u.)
hω(meV)
CA B
Figure 4.8: (A) This is the dispersion of the mode centered at (π, π) mea-sured by inelastic neutron scattering on heavily underdoped Y Ba2Cu3O6+x
below TC (Figure adapted from Ref. [26]). (B) This is the dispersion ofthe same mode, measured on a nearly optimally-doped Y Ba2Cu3O6+x aboveTC (Figure adapted from Ref. [27]). Allthough there is a peak in the data,the features have become too broad to identify a dispersion. (C) This is thedispersion of the mode centered at (π, π) measured by inelastic neutron scat-tering on La2−xBaxCuO4 above TC (Figure adapted from Ref. [28]). In otherLanthanum-based compounds, there is no appreciable change in dispersionabove and below TC . [29]
61
[29, 27, 26] In the pseudogap state, the features near the antiferromagnetic points persist,
[28] although their dispersing nature becomes washed out in some materials (Fig. 4.8b &
c). [27] Because the spin ground state is not known in general, as there is no elastic neutron
scattering signal in the non-Lanthanum-based compounds, the inelastic neutron scattering
data cannot be directly interpreted as proof of spin waves, but can be interpreted as general
evidence for fluctuating antiferromagnetic spin correlations. However, no clear connection
has as yet been made between the inelastic neutron data and the pseudogap temperature or
energy scales in an experiment.
4.1.3 Theoretical Ideas
Innumerable theoretical attempts have been made to understand the pseudogap in the den-
sity of states, and, more broadly, the behavior of the HTS in the pseudogap regime. At-
tempting to segregate them into well-defined groups is a somewhat artificial procedure, as
the theories often attempt to explain both the experimental data supporting precursor su-
perconductivity and fluctuating spin order, and tend to borrow ideas from one another.
However, for the purposes of discussion, they can be roughly divided into two categories:
those using the superconducting state as their starting point, and those using exotic ordered
states as their starting point.
One set of theories ascribes the pseudogap in the density of states to preformed Cooper
pairs existing above the temperature (TC) where they become phase-coherent. Such behavior
is not unusual for planar materials. True off-diagonal long-range order should, thermody-
namically, only occur at T = 0 in two- dimensional systems. Global phase transitions in
quasi-two dimensional materials occur when thermal fluctuations become as large as (often
hidden) interactions that make the system not perfectly two dimensional, like out-of-plane
interactions. Meanwhile, local correlations can have significantly higher energy scales associ-
ated with them because they are associated with in-plane interactions. In the HTS, however,
62
both the phase stiffness energy and the anomalously large Nernst signal disappear at temper-
atures below where the pseudogap in the density of states does. [18, 19] Still, the pseudogap
could well be tied to a pairing energy scale, the Nernst and optical conductivity temper-
atures to the establishment of fluctuating (in-plane) phase coherence between these pairs,
and the superconducting transition to the global phase coherence of these fluctuating super-
conducting patches. The precursor superconductivity scenario for the HTS is qualitatively
analogous to the behavior of the antiferromagnetic insulator at low doping. [35, 20] Here,
three-dimensional antiferromagnetic order survives up to a temperature TN , which is associ-
ated with the in-plane magnetic anisotropy and the out-of-plane magnetic exchange energy.
Thermal fluctuations become a limiting factor for in-plane correlations only at tempertures
well above TN . Finally, at experimentally unobtainable temperatures, thermal fluctuations
reach the strength of the nearest neighbor exchange energy and the local antiferromagnetic
’pairing’ of nearest neighbor spins vanishes. However, there is as yet no solid experimental
proof that the observed pseudogap is actually connected with Cooper pairing.
Other proposals seeking to explain the pseudogap in the density of states have focused
on the possibility that some form of ordering is responsible for the measured behavior of the
underdoped cuprates above TC . Pseudogap-like depressions in the density of states associated
with charge ordering have been seen in other layered metal oxides, such as K0.9Mo6O17 [36],
and with both charge and spin order in the dichalcogenides [37].
One set of theoretical efforts proposing exotic order in the pseudogap regime treats the
microscopic details of doping an antiferromagnetic insulator into the pseudogap regime. In
the undoped insulator, which has one electron per Cu atom, Coulomb repulsion localizes
electrons on the Cu sites of the CuO2 plane. These electrons can lower their total energy
by being allowed to hop from one site to another. Because this would be prevented by the
Pauli Exclusion Principle for aligned nearest neighbor spins, the undoped insulator develops
antiferromagnetic spin alignment. (Fig. 4.9a). One proposal suggests that doping this
63
antiferromagnetic lattice is best thought of as turning the lattice into a liquid of spin pairs
(Fig. 4.9b). [38] Although it is known that the undoped lattice has long-range (crystalline)
antiferromagnetic order, this order is destroyed with the removal of a very small concentration
of spins (just one in 25). However, the spin at a given site can be thought of as bound to a
nearest neighbor, forming a spin singlet. Because it can be equivalently bonded to any one
of its nearest neighbors in the antiferromagnetic state, a situation analogous to the single-
bond, double-bond resonance of the Carbon atoms in a Benzene ring, this state is called
the resonant valence bond state (RVB). The RVB scenario suggests that the destruction
of antiferromagnetic order arises from the ability of the spin pairs to slide around, as if
comprising a liquid. Theoretical works have shown that the hole-doped spin liquid state
has the remarkable property that charges and spins can be treated separately. [39] The
pseudogap would then arise from the energy cost of having to remove a real electron, which
requires breaking apart a spin pair. In addition, these spin pairs have been conjectured to be
analogous to the Cooper pairs in BCS superconductivity. [39] This is appealing as well, as
the shape of the superconducting dome in the phase diagram would then be limited by the
low number of charged carriers for underdoping (despite a growing pair potential for spins),
and by the shrinking pair potential for spin pairing with overdoping (despite a growth in the
number of holes).
Another prominent proposal, that of ordered circulating currents, has come out of mi-
croscopic ideas developed in understanding the resonant valence bond scenario. The spin
liquid has instabilities to various kinds of ordered states, [40, 41, 42] including some involving
currents circulating either between the Cu atoms [43], or between the Cu and O atoms (Fig.
4.10a). [44] An experimental hallmark of such a state is that it breaks time-reversal sym-
metry. Signatures of this broken symmetry might have been seen in a recent photoemission
experiment, although the data remains highly controversial. [45] Moreover, no microscopic
observation of circulating current order has been made. In both scenarios, the circulating
64
A B
Figure 4.9: (A) This is a schematic diagram of the antiferromagnetic latticepresent in the undoped insulator. (B) This is a schematic diagram of thespin-pairs in the resonant valence bond picture. Removal of less than one spin(grey) in 25 is enough to destroy the crystalline antiferromagnetic state, andallow the spin singlets to form a liquid. Figures (A) and (B) have been adaptedfrom Ref. [32].
current order competes with superconductivity. Thus, the suppression of TC at low dopings
arises from the increase in size of the circulating current order parameter, with the suppres-
sion of TC at high dopings arising from the decrease in the size of the order parameter for
superconducting order.
The concept of spin-charge separation, often discussed in the context of the resonant
valence bond scenario, has played a prominent role in another ordering proposal for the
pseudogap that also derives from the microscopic physics of doping an antiferromagnetic
insulator- the so-called stripe model. [47, 48] The stripe model suggests that holes added to
the parent antiferromagnet organize into one-dimensional spinless charge structures separat-
ing antiferromagnetic domains. [30] The separation between neighboring stripes, and hence
the size of the antiferromagnetic domains, is then directly related to the doping level of
the sample. This evolution of the antiferromagnetic spin structure is consistent with elastic
neutron scattering experiments on the Lanthanum family of the HTS, which reveal that the
size of antiferromagnetic domains shrinks as holes are added (Fig. 4.7). [25]
Another set of theories have examined the doped antiferromagnetic insulator from a more
global perspective, not starting from microscopic details. They suggest that the inelastic
65
Figure 4.10: Two proposed types of circulating current order for the HTSare shown. On the left is the so-called d-density wave state, which involvescurrents circulating between Cu sites (from Ref. [43]). On the right is a stateproposed by Varma, where currents circulate between a Cu site and the O itsbonded to (from Ref. [46]).
neutron data of Figure 4.8 indicates that fluctuating spin order survives even when the
undoped insulator is doped across the antiferromagnetic phase boundary. These theories
postulate that fluctuating spin order is responsible for the pseudogap behavior, [49, 50]
and competes with superconducting order, [51, 52] resulting in the suppression of TC for
underdoped samples.
The final proposal we will discuss here has culled together many ideas attributed to
the other models discussed here- the so-called SO(5) theory. This theory proposes that
superconducting order and antiferromagnetic order are degenerate, and can be described by
a single order parameter with five degrees of freedom- the real and imaginary parts of the
superconducting order parameter, and the three orthogonal degrees of freedom associated
with a spin vector. [53] At a high temperature scale, spins pair to form singlets, giving this
five dimensional order parameter a finite amplitude. Meanwhile, at lower temperatures, the
degeneracy between the superconducting and antiferromagnetic degrees of freedom can be
lifted by relatively small energy scales, which selects a phase for the order parameter and
determines whether the pairs have antiferromagnetic and/or superconducting order. The
antiferromagnetic state is essentially a crystalline lattice of Cooper pairs, which, when holes
66
are doped into the system, melts into a superfluid of Cooper pairs at low temperatures.
The pseudogap in the density of states is then associated with the initial formation of the
pairs (the finite amplitude SO(5) order parameter, with undefined phase). This amplitude
decreases with the addition of holes, consistent with the behavior of the pseudogap in the
density of states. The shape of the superconducting dome is limited by increased fluctuations
due to the proximity of antiferromagntism for low doping, and the decrease in pair amplitude
at overdoping. Although starting from global considerations, the microscopic picture that
emerges for the pseudogap regime is that of spin singlets that increasingly freeze into a
crystalline lattice as doping decreases. [54]
To date, no experimental technique has made a definitive identification of the physics
underlying the pseudogap regime. Although the proposals outlined here have been grouped
into separate categories, many share common features. All the proposals share the common
feature that the characteristic length scales are on the order of the lattice constant. The
precursor superconductivity scenario has a characteristic length scale set by the supercon-
ducting coherence length, just 16A in-plane for the HTS. [55] The scenarios based on various
proposed exotic order have characteristic length scales on the order of the lattice constant.
Because STM can measure the electronic density of states as a function of location on these
length scales, and directly measures the pseudogap, it could be an ideal tool to study pseudo-
gap physics. In this chapter, we use STM to look for signs of electronic ordering on a sample
in the pseudogap regime for the first time. [56] We find a dispersionless, incommensurate
standing wave pattern in the spatially resolved density of states at fixed energies inside the
pseudogap. The energy-independent wavelength of these patterns indicate that they cannot
be the result of scattering interference, and suggest they are the signature of some form of
local ordering in the pseudogap regime. Finally, we review how the measured patterns relate
to some of the theoretical proposals for ordering in the pseudogap regime.
67
4.2 Results
To search for signs of local electronic ordering in the pseudogap regime of the HTS, we used a
home-built variable temperature STM to map the spatial dependence of the density of states
at several energies. A slightly underdoped single crystal Bi2Sr2CaCu2O8+δ sample with 0.6%
Zn impurities (TC = 80K) was first cleaved in ultra-high vacuum at room temperature, then
thermalized in the STM at T = 100K. Topographic images of the surface (Fig. 4.11b)
show an atomic corrugation and b-axis superlattice distortion similar to that seen at low
temperatures (Fig. 3.4a). A tunneling conductance curve taken at an arbitrary position
on the sample (Fig. 4.11a) shows a pseudogap of ∆ ≈ 35 − 40mV centered at the Fermi
energy, in agreement with previous measurements made on slightly underdoped samples
above TC . [1] We will focus on the spatial distribution of electronic states at fixed energies
inside this pseudogap (Fig. 4.11c-f) taken simultaneously with the topograph in Figure
4.11b. The spatially resolved density of states at high energies shows the expected structural
features- the atomic corrugation and the b-axis superlattice distortion. In addition, there
are bright and dark patches in the local density of states, thought to originate from local
dopant inhomogeneity. [57] Upon lowering the energy below the pseudogap energy scale, a
CuO bond-oriented modulation appears, and becomes more prominent upon decreasing the
energy to the Fermi energy. Similar results have also been acquired on slightly underdoped
samples containing no Zn (Fig. 4.12).
To characterize these new modulations, we performed a two-dimensional Fourier analysis
of the conductance maps. The Fourier transformed density of states maps (Fig. 4.13a)
typically contain three features (shown schematically in Fig. 4.13b): one, a set of four peaks
along the 〈0, π〉 directions corresponding to the atomic corrugation (A); two, a set of two
peaks along the 〈π, π〉 directions corresponding to the b-axis superlattice distortion (S); and
three, a set of four sharp peaks along the 〈0, π〉 directions (the Cu − O bond direction)
68
150 pS
35 pS
10 n
m
dI/dV
(nS
)
Voltage (mV)
0.3
0.1
0.2
00−150 150
A B
C D
E F
6 mV12 mV
24 mV41 mV
Figure 4.11: Underdoped Bi2Sr2CaCu2O8+δ samples with 0.6%Zn impuri-ties (TC = 80K) were investigated using an STM operating at T = 100K.(A) STM topographic data taken on a representative 450A × 195A field ofview with a tunneling current I = 40pA and a sample bias of −150mV re-semble topographic maps taken at low temperatures. (B) A representativeSTM conductance spectrum taken at a fixed position as a function of energy(Rj = | − 150mV |/40pA = 3.75GΩ) exhibits a pseudogap of ≈ 35 − 40mVcentered at the Fermi energy. Spatially resolved conductance maps were ac-quired simultaneously with the topograph in (A) at (C) 41mV, (D) 24mV,(E) 12mV, and (F) 6mV. All STM conductance spectra were taken using astandard ac lockin technique with a bias modulation of 4mVrms. Figures havebeen adapted from Ref. [56]
69
A B C
Figure 4.12: Underdoped Bi2Sr2CaCu2O8+δ samples with no Zn impuri-ties (TC = 85K) were shown to produce qualitatively similar STM data atT = 100K as samples with Zn impurities (Fig. 4.11). (A) An STM topo-graph of a typical region taken with I = 40pA and V = −150mV showsthe expected atomic corrugation and superlattice distortion. Simultaneouslyacquired spatially-resolved conductance maps contain primarily structural fea-tures at (B) 30mV , and the emergence of a new bond-oriented modulation at(C) 10mV .
corresponding to the new modulation (Q). The new modulation behaves very differently
than the structural peaks. To see this, the intensities of the peaks were first scaled to their
value at 41mV (Q, 70.3pS; S, 227.7pS; A, 35.5pS), and then tracked as a function of energy
(Fig. 4.13c). The scaled amplitudes of the peaks associated with structural features (A and
S) decrease inside the pseudogap as the energy approaches the Fermi energy. This mirrors
the overall drop of the density of states inside the pseudogap, and is consistent with the
behavior of structural features in STM density of states maps taken on other surfaces. In
contrast, the new modulation (Q) only appears for energies inside the pseudogap, and its
scaled amplitude grows as the energy approaches the Fermi energy, suggesting that the new
modulation has an electronic origin, and not a structural one.
Analyzing the structure of the new modulation Q in Fourier transform space reveals
precise quantitative information about the real-space structure of the modulations. The
locations of the Q and A peaks can be tracked by taking two-pixel-averaged line-cuts of the
FFT, shown as the dashed line in Figure 4.13b. From determining their peaks’ location (Fig.
4.13d), we find that these modulations have a real-space incommensurate period of 4.7a0,
where a0 = 3.8A is the nearest neighbor Cu − Cu distance, in these slightly underdoped
70
A
k (2π/ao)
Fo
urie
r a
mp
litu
de
(p
S)
D Q −15 mv
0 mv
15 mv
100
50
0
150
0 0.2 0.4 0.6 0.8 1
A
Q
S
B <0,π
><π,π>
0
0.2
0.4
0.6
0.8
1
1.2
1.4
1.6
2
2.1
2.2
2.3
0 20 40 60 80 100
<π-0> peak (Q)
b-axis peak (S)
atomic peak (A)
DO
S (n
S)
DOS
No
rma
lize
d p
ea
k in
ten
sity
Voltage (mV)
<0,π
><π,π>
A
C
Figure 4.13: (A) The Fast Fourier Transform of a conductance map (I = 40pAand V = −150mV ) acquired on a 380A × 380A area with 200 × 200 pixels at15mV has four basic components. (B) These components are labeled in thisschematic diagram as A for the atomic peaks, S for the superlattice distortionpeaks, and Q for the new bond-oriented modulation. The broad peak in thecenter (unlabeled) corresponds to long-wavelength changes in conductance as-sociated with dopant inhomogeneity in the sample. (C) The relative intensityof the structural peaks (A and S) decreases inside the pseudogap, while that ofthe new modulation (Q) increases. Also shown is the energy-resolved densityof states (green). (D) Shown are line-cuts from tranformed STM data acquiredsimultaneously at 15mV (shown in (A)), 0mV and -15mV. Notably, the po-sition of the nearly resolution-limited Q peaks does not shift as a function ofenergy.
71
samples containing Zn. Importantly, the positions of the Q peaks does not shift for all the
energies where these peaks are seen on a given sample to within our experimental resolution
of ±0.2a0. Measurements on different samples, including those not containing Zn, find
a variation in this period for a given sample between 4.5a0 and 4.8a0. This supports an
electronic origin to the modulations as well, since the atomic lattice peak (A) and the b-axis
supermodulation peak (S) had the same wavevector in all the samples. Finally, the narrow
width of the Q peaks indicates that the modulations have a correlation length of about 4−5
periods, which is corroborated by the somewhat disordered appearance of the modulations
in the real-space data. The appearance of a non-dispersing, incommensurate standing wave
pattern in the real-space density of states inside the pseudogap is the main experimental
result of this chapter.
The distinguishing feature of the modulations in the pseudogap regime is their energy-
independent wavelength. Although they appear similar to the modulations in the real-
space density of states measured by STM in the superconducting state (Chapter 3), the
modulations in the superconducting state were found to disperse. A potentially simple
explanation for the lack of dispersion seen in the pseudogap regime is that the patterns do
have an underlying dispersion, but the large level of energy broadening (±16mV for the
measurements presented here) would simply average over a range of wavevectors specified
by the underlying dispersion. Although this could produce modulations with a seemingly
fixed wavevector, the energy broadening would also result in a broadening of features in the
Fourier transforms of the data. We measure the modulations to have a wavelength of 4.7a0
with an uncertainty of ±0.2a0, where the uncertainty is due to the observed width of the Q
peaks. Thus we need to identify whether some mechanism can produce either dispersionless
modulations in the density of states, or modulations that disperse less than ±0.2a0 over the
range of energies (−15mV to ≈ 40mV ) where we find the modulated patterns.
72
4.3 Scattering Interference
Recall that the modulated patterns found in the superconducting state were thought to arise
from scattering interference, as described in Chapter 3. [58, 59, 60, 61, 60, 62, 63] We can
attempt to extend the scattering interference scenario into the pseudogap regime to describe
the dispersionless modulations seen here. [64] Although angle-resolved photoemission spec-
troscopy does not find well-defined quasiparticles for T > TC , the electronic states at a given
energy can still be thought to have a broad range of available wavevectors, and should be
able to scatter elastically between them. In the Born approximation, scattering interference
introduces a correction
δn(~r, ω) = − 1
πIm
∫
d2~r1G0(~r − ~r1, ω)V (~r1, ω)G0(~r1 − ~r, ω) (4.1)
to the density of states, where G0 is the single particle Green function, and V is a weak,
finite-range scattering potential. [65, 60, 66, 67, 63, 68] The Fourier transform of the Born
correction δn(~q, ω) = − 1πImV (~q, ω)Λ(~q, ω) separates into a part
Λ(~q, ω) =∫
d2~kG0(~k, ω)G0(~k + ~q, ω) (4.2)
that contains all the wave interference information, and a part
V (~q, ω) =∫
d2~xe−i~q·~xV (~x, ω) (4.3)
that acts like a static structure factor. [66] For now, we assume that the structure factor
acts like an all-pass filter, and the density of states correction depends only on the wave
interference term Λ.
Before addressing the various electronic states proposed for the pseudogap regime, we
treat the case of the Fermi liquid normal state for didactic purposes. Assuming that the
electronic state of the sample can be described as a metal, the Green function is given as
G0(~k, ω) = (ω + iδ − ε~k)−1 (4.4)
73
where δ is a broadening term, and ε~k is the Bi2Sr2CaCu2O8+x band structure from ARPES
[69] for slightly underdoped (µ = 120.5meV , corresponding to a doping of x = .12)) samples,
[66] The resultant correction to the density of states contains dispersing caustics, in contract
to the sharp peaks in ~q-space for the superconducting state (Fig. 4.14a&b). The equivalence
of all points in ~k-space in the Green function leads directly to an absence of sharp peaks
in ~q-space. Furthermore, the dispersion is solely the consequence of the band structure
(Fig. 4.14c). Although the Fermi liquid picture helps to illustrate the effect of the band
structure on scattering interference, it is not directly applicable to the STM data measured
in the pseudogap regime. We must still account for the two key phenomenological features
of the pseudogap regime measured by photoemission: the pseudogap in the density of states
n(~k, ω) = − 1πImG0(~k, ω), and the ill-defined nature of the electrons as a function of both
energy and momentum.
Extending the scattering interference scenario into the pseudogap regime requires choos-
ing a Green function with which to model the enigmatic pseudogap electronic state. Recall
that in the pseudogap regime, angle-resolved photoemission spectroscopy (ARPES) finds
that electronic states form extended arcs centered along the 〈π, π〉 directions in ~k-space at
the Fermi energy. [70] As the energy approaches the pseudogap energy scale, these arcs
extend towards the Brillouin zone boundary, and form a closed contour for ω > ∆ (Fig.
4.15a). Finally, the ARPES lineshape is quite broad as a function of both energy and mo-
mentum, calling into question whether true quasiparticles exist at all. [14] Features of the
measured behavior have been modeled by numerous theoretical works, which have typically
either followed a phenomenological approach, or proposed exotic electronic states. We start
by focusing on the former.
Norman et al. [71] have captured many details of the ARPES data with the Green
function
G0(~k, ω) = (ω − ε~k − Σ~k)−1 (4.5)
74
A B
DC
q=(0,0) (0,π)
(π,π)
k=(0,0) (π/a,0)
(π/a,π/a)
2
3
4
5
6
7
8
-40 -20 0 20 40Energy (mV)
Wav
ele
ng
th (
a0)
Figure 4.14: We calculated the power spectrum of δn(~q, ω) using the Fermiliquid Green function with δ = 1mV . Shown are the ~q-space patterns in thefirst Brillouin zone at ω = 0 (A), and ω = 32mV (B). The dispersion of thefeature along the 〈0, π〉 direction, indicated by the arrows in (A)&(B) is plottedin (C) as a solid line. Also shown are is the dispersionless STM data takenat T = 100 > TC on a slightly underdoped sample (closed circles). [56] (D)The reason for the dispersion is that the contours of constant electron energy(dark blue, 0mV ; light blue, 32mV ) change shape in ~k-space, as dictated bythe band structure.
75
where Σ(~k, ω) = −iΓ1 +∆2~k/(ω + ε~k + iΓ0) is the self-energy, Γ1 is a single particle scattering
rate, Γ0 is a measure of decoherence, and ∆~k is a gap function of amplitude ∆ = 45mV with
a ~k dependence which matches the Fermi arcs. The most promising aspect of the Fermi arc
model is that, for unknown reasons, the wavevector that joins the tips of the Fermi arcs at
the Fermi energy has the same wavelength (4.6a0) as the patterns seen by STM.
The scattering interference patterns calculated using this phenomenologically-motivated
Green functions are unlike the measured STM data (Fig. 4.15). First, the calculated pat-
terns contain caustics, while the measured patterns contain sharp peaks along the 〈0, π〉
directions. This can be addressed by assuming, as has been suggested in the literature,
that tunneling occurs preferentially along the 〈0, π〉 directions. Second, the features seen
in the calculation along the 〈0, π〉 direction disperse, unlike the dispersionless peaks seen in
the data. As indicated by the arrow, which is of the same length in both Fig. 4.15c&d,
there is a measurable dispersion for the slowest dispersing mode along the 〈0, π〉 direction.
The calculated dispersion resembles the dispersion calculated for the superconducting state
(see Fig. 4.16e) because the strongest wavevector for scattering interference is of a similar
length at the gap energy for both cases, but is shorter in the pseudogap regime at the Fermi
energy. However, the shallower dispersion, which results in a change in wavelength between
7a0 at the Fermi energy and 4.6a0 at the pseudogap energy, should still be easily resolved
in the data, which sees a fixed wavelength of 4.7 ± 0.2a0 over the entire energy range (Fig.
4.15d). [56] Unphysically small values of Γ0 = 1mV and Γ1 = 1mV were chosen to produce
sharper features in the calculations shown. However, larger (≈ 10mV ) values of Γ0 lead to
the gradual suppression of the observed mode in favor of a mode that disperses similar to
the band structure (Fig. 4.14). Larger values of Γ1 yield pictures that average over some
part of the dispersion for a particular mode, leading to features significantly broader than
the sharp peaks seen in the data.
We next calculated scattering interference patterns using various exotic electronic states
76
B
2
3
4
5
6
-40 -20 0 20 40
Wav
ele
ng
th (
a0)
Energy (mV)
E
k=(0,0) (π/a,0)
(π/a,π/a)
A C
q=(0,0) (0,π)
(π,π)D
q=(0,0) (0,π)
(π,π)
Figure 4.15: (A) This is a model representation of the electronic density of
states in ~k-space measured by ARPES for underdoped samples above TC .Shown are the curves of constant electron energy for ω = 0mV (light blue)and ω = 32mV (dark blue), and the charcteristic scattering vectors ~q along the〈0, π〉 direction at both energies (light red, dark red). The power spectrum ofδn(~q, ω) calculated using a Fermi arc Green function is shown here at ω = 0mV(C) and ω = 32mV (D).
77
2
3
4
5
6
7
-40 -20 0 20 40Energy (mV)
Wav
ele
ng
th (
a0)
A
EDC
B
q=(0,0) (0,π)
(π,π)
Figure 4.16: The power spectrum of δn(~q, ω) calculated using the preformedpairs Green function of Chen et al. [72] is shown here at ω = 32mV forΓ0 = 1mV (A) and Γ0 = 10mV (B). The power spectrum of a QED3 Greenfunction [65] is shown here at ω = 32mV for η = 0.4 (C) and η = 1.2 (D). Thevalue of the energy gap was chosen to be ∆ = 45mV for both calculations.(E) The dispersion of the mode identified by the arrow in both (A) and (C) isshown here as the grey line. The dispersion of the mode in (B) is also shown,and matches the dispersion of the band structure. Finally, the solid circlesdenote the dispersion measured by STM in the pseudogap regime. [56]
78
proposed for the pseudogap regime. Any number of proposals have been put forth for the
pseudogap electronic state, but we will focus on four of these exotic Green functions- two
based on precursor superconductivity, and two based on other forms of order. Chen et al.
[72] have proposed that the pseudogap is due to preformed pairs, and can be described by a
modified Fermi arc Green function that has a d-wave energy gap. The calculated scattering
interference pictures based on this Green function resemble those of the superconducting
state for Γ0 small, and a broad version of the Fermi liquid state for Γ0 large (Fig. 4.16a&b).
In either limit, the patterns disperse, and should be resolved easily by STM (Fig. 4.16e).
Pereg-Barnea and Franz [65] propose a different Green function to describe preformed Cooper
pairs,
G0(~k, ω) = (ω + ε~k)/(ω2 − ε2~k− ∆2
~k)1−η/2, (4.6)
with η serving as a tunable parameter controlling the level of decoherence. As seen in Figure
4.16c&e, the scattering interference patterns generated using this Green function produce a
dispersion similar to the superconducting state (Chapter 3) for η < 0.5. Larger values of η
produce increasingly diffuse features in the calculated density of states correction until, by
η = 1.0, no features are discernable in ~q-space whatsoever (Fig. 4.16d).
Scattering interference calculations based on other proposed exotic electronic states in the
pseudogap regime also contain an unacceptable amount of dispersion. In Figure 4.17a&b, we
show scattering interference patterns for the marginal Fermi liquid scenario with circulating
current order. [73] The Green Function for this phase,
G0(~k, ω) = (ω − ε~k ± D(~k) − Σ~k)−1 (4.7)
with the plus (minus) sign for k > kF (k < kF ) , Σ(~k, ω) = λω log x/ωc+iπx/ cosh (D(~k)/x),
x = (ω2 + π2T 2)1/2 and D(~k) = D0(cos(kxa) − cos(kya))2, produces patterns that disperse
in a fashion similar to the band structure (Fig. 4.17a&b). Thus, it also fails to match the
dispersionless feature seen in STM in the pseudogap regime (Fig. 4.17d). It is replaced by a
79
faster dispersing mode at energies |ω| < 2D0 for finite values of D0, although the large energy
broadening of the Green function washes this mode out too much to track its dispersion. The
last proposal we illustrate in this chapter is the scattering interference patterns calculated
using the d-density wave proposal (DDW). [65, 74] As discussed in the literature [65], the
density of states correction arising from scattering interference in the DDW proposal is given
by
δn(~q, ω) = − 1
πImV (~q, ω)
∫
d2~k G0(~k, ω)G0(~k + ~q, ω)
+F0(~k, ω)F0(~k + ~q, ω)
+G0(~k, ω)F0(~k + ~q, ω)
+F0(~k, ω)G0(~k + ~q, ω)
(4.8)
where
G0(~k, ω) = (ω − ∆µ − ε~k+ ~Q)/((ω − ∆µ − ε~k)(ω − ∆µ − ε~k+ ~Q) − D2~k), (4.9)
F0(~k, ω) = D~k/((ω − ∆µ − ε~k)(ω − ∆µ − ε~k+ ~Q) − D2~k), (4.10)
D~k = D0(cos kx−cos ky)/2 is the DDW gap, ∆µ is a chemical potential shift, and ~Q = (π, π).
As can be seen in Figure 4.17c&d, the patterns disperse through a range of wavelengths
(∆λ = 0.8a0 between −15mV and 35mV ), which should be easily resolved in the STM
experiment (maximum ∆λ = 0.4a0 over the same range of energies).
The inability of the proposed electronic states to produce dispersionless scattering inter-
ference density of states corrections points to a fundamental problem: any Green function
will result in a wave contribution to Born scattering that disperses. This happens because
we have placed too many restrictions on the Green function. The occupied density of states
(the imaginary part of the Green function times the Fermi function) must disperse in order
to match ARPES measurements. On the other hand, the density of states correction from
80
scattering interference, which is just the imaginary part of the Green function convoluted
with itself, cannot disperse if it is to match the STM data in the pseudogap regime. In
order for scattering interference to simultaneously match both the ARPES and STM data,
the structure factor, assumed to be an all-pass filter until now, could be chosen to pass only
contributions near ~q = 2πa0
(0, 1/4.7). [66] However, this corresponds to an incommensurate,
bond-oriented square lattice of scattering centers in real-space. Although this can be justified
on various physical grounds, [67, 68] it amounts to an ad hoc assumption that some form of
ordering exists in the material, and electrons are scattering off the ordered scattering sites
to produce dispersionless patterns.
These qualitative observations have been made by calculating density of states corrections
in the Born approximation (weak scattering). In the Born approximation, we have considered
only the first order correction to the Green function arising from elastic scattering: G =
G0 +G0V G0, where G0 is the unperturbed Green function, and V is the scattering potential.
However, the arguments also apply in the strong scattering limit, where we consider the
same type of correction to all orders: G = G0 + G0V G0 + G0V G0V G0 + .... Starting from a
more general form for elastic scattering,
δn(~r, ω) =∫
d2~kG0(~k, ω)T (~k,~k + ~q, ω)G0(~k + ~q, ω), (4.11)
where
T (~k,~k + ~q) = V (~k, ~k + q) +∫
d2~k′V (~k, ~k′)G0(~k′)T (~k′, ~k + ~q). (4.12)
The behavior of δn will be determined by the convolution of G0 with itself if T has no poles
at a given energy. In order for T to have poles as a function of ~k and ~k + ~q, V must contain
poles itself. Thus, in order to overcome the dispersion in δn arising from G0, we must assume
that V has poles in it, corresponding to an ordered real-space array of scattering sites. We
are left to conclude that the modulations seen in the density of states measured by STM in
the pseudogap regime simply cannot be described by scattering interference.
81
q=(0,0) (0,π)
(π,π)A B
C
2
3
4
5
6
-40 -20 0 20 40
Wav
ele
ng
th (
a0)
Energy (mV)
D
Figure 4.17: The power spectrum of δn(~q, ω) was calculated with the circulat-ing current Green function using ∆0 = 10mV (A) and ∆ = 0 (B), shown hereat ω = 16mV . Values of λ = .27, T = 10mV , and ωc = 500mV were usedin both calculations. The mode pointed to by the arrow in (B) disperses likethe Fermi liquid case. (C) The power spectrum of δn(~q, ω) calculated usingthe d-density wave formalism [74, 65] is shown here at ω = 16mV . The valueof the energy gap was chosen to be ∆ = 45mV , and the chemical potentialshift was chosen to be ∆µ = 40mV . (D) The dispersion of the 〈0, π〉 modescalculated using the circulating current Green function with D0 = 0 is shownhere in grey. Also shown is the slowest dispersing 〈0, π〉 mode for the DDWGreen function (black), and the faster dispersing mode identified by Ref. [75](open circles). Finally, the solid circles denote the dispersion measured bySTM in the pseudogap regime. [56]
82
4.4 Local Order
The most obvious explanation for the modulated patterns seen by STM in the pseudogap
regime would be that they result from some form of ordering. In general, the patterns seen
in the STM data could result from local charge ordering itself, from scattering off a static
ordered state, or from scattering off fluctuating order pinned by defects in the sample. One
obvious candidate for the modulations is charge stripes, as the patterns seen in STM have a
periodicity close to that expected from the stripe ordering scenario. However, the patterns
seen here naively appear to be two-dimensional, which would require one-dimensional stripes
to be either fluctuating or highly disordered in two orthogonal directions.
Another promising explanation is the local ordering of spins predicted by numerous the-
ories. [47, 53, 51, 52] These theoretical predictions have often been found to be consistent
with either the elastic spin structure or the inelastic spin fluctuations measured by neu-
tron scattering (or both). Recent theoretical work has attempted to connect the real-space
periodicity seen in STM conductance measurements and the neutron scattering data. [67]
This work postulates that spin fluctuations with wavevector 2π/λ can be pinned by de-
fects. Scattering interference off this pinned spin wave structure leads to electronic density
of states modulations with wavevector 4π/λ. In the superconducting state, the agreement
between the wavevector measured by neutron scattering as a function of energy, and half the
wavevector of the DOS modulations measured by STM, is quite remarkable (Fig. 4.18). [76]
However, neutron scattering measures the same dispersion in the Lanthanum compounds
above TC (Fig. 4.18b), [76] and only very broad features in other compounds (Fig. 4.8c).
[27] In contrast, STM sees a comparatively sharp, non-dispersing feature in the electronic
density of states. What’s more, there is no measured correlation between the spin informa-
tion measured in neutron scattering and the pseudogap energy or temperature scales, while
there is a correlation between the appearance of the dispersionless modulations seen here
83
A B
Figure 4.18: (A) The wavevector of the spin wave measured in inelastic neutronscattering (K is the distance away from (π, π) measured in reciprocal latticeunits) is plotted as a function of energy for optimally doped LSCO (diamondsand sqaures) and underdoped YBCO (circles) in the superconducting state.Also shown as the red symbols is half the wavevector of the bond-orientedmodulations measured by STM in the superconducting state. [58] (B) Shownis the wavevector of the spin wave (δ, measured relative to the (π, π) point) asa function of energy for samples in the superconducting (blue) and pseudogap(red) states. The data was acquired on optimally doped LSCO (squares anddiamonds) and underdoped LSCO (triangles). In contrast, the modulated pat-terns seen in STM data are non-dispersing in the pseudogap regime. (Figureadapted from Ref. [76]).
with STM and the pseudogap energy scale (Fig. 4.13).
The STM data presented here could also be compatible with precursor superconductivity
scenarios for the pseudogap. [77] Recall that these proposals link the pseudogap to Cooper
pair correlations above TC . Theoretical works have shown that a pair liquid with a fluc-
tuating phase, consistent with the loss of global superconducting order and the persistence
of superconducting fluctuations, can form modulated patterns in the density of states. [65]
However, these modulations should disperse in a manner similar to the quasiparticle scat-
tering picture. Alternatively, proposals have been made that hole pairs can localize into a
disordered lattice. [78, 79] These proposals, which combine elements of the ordering sce-
nario with precursor superconductivity, yield a density of states modulation with the right
periodicity, and whose appearance correlates with the pseudogap energy scale.
Accompanying the issue of what drives the local electronic ordering is whether the local
84
order competes with, coexists with, or promotes superconductivity. Here, we have shown that
in-gap states form dispersionless periodic modulations when superconductivity is suppressed
by raising temperature above TC . In an earlier study, STM was used to examine the in-gap
states when superconductivity is suppressed at low temperatures by applying a magnetic
field. Near vortices, where magnetic flux threads the sample, STM sees what appears to
be a low-temperature version of the pseudogap in the density of states (Fig. 4.19a). [80]
A modulated spatial pattern with a wavelength of ≈ 4.3 ± 0.5a0 is found upon integrating
the measured density of states between the Fermi energy and 10mV (Fig. 4.19b). [81]
Unfortunately, the lack of energy- resolved data near vortices prevents determination of
the amount of dispersion for these patterns. Another recent study has searched for spatial
modulations in the density of states on samples where superconductivity has been suppressed
by heavy underdoping. [82] Although the samples used were at superconducting dopings
(δ = 0.1 and above), they find that the density of states has a large (δ ≈ 100mV ) pseudogap
(Fig. 4.19c), and shows little evidence for any feature associated with superconductivity.
The spatially-resolved density of states maps of the surface show a dispersionless modulated
pattern, with a commensurate period of 4a0 (Fig. 4.19d). In all three experiments, whenever
superconductivity is suppressed, whether by temperature, magnetic field, or doping, and
a pseudogap is seen in the electronic density of states, STM finds similar dispersionless
modulations in the local density of states.
It must be emphasized that more experiments must be done before concluding that the
modulated density of states patterns actually cause the pseudogap in the density of states.
First, it remains to be seen whether the appearance of the patterns correlates with the
appearance of the pseudogap throughout the phase diagram. Although we show here that
this correlation exists for slightly underdoped samples at T = 100K, and the correlation
has also been demonstrated for heavily underdoped samples at T = 0.1K, experiments
remain to be done on samples which do not exhibit a pseudogap, and it remains to be
85
C DA B
Figure 4.19: (A) This is the energy-resolved density of states measured ata vortex core on a nearly optimally doped Bi2Sr2CaCu2O8+δ sample at lowtemperatures in an applied field(figure adapted from [80]). (B) The spatially-resolved, energy-integrated density of states of a typical region also show abond-oriented modulation near the vortex cores with a period of 4.3a0 (figureadapted from Ref. [81]). (C) This is the energy-resolved density of states inheavily underdoped (x = 0.12) Ca2−xNaxCuO2Cl2 measured at low tempera-tures. (D) The spatially resolved density of states on the same surface, shownhere at 24mV , shows a bond-oriented modulation with a period of 4a0. Boththese figures have been adapted from Ref. [82].
seen whether it is possible to find samples with a pseudogap and no modulated patterns.
Independent of whether the appearance of the patterns is correlated with the presence of
a pseudogap, it could still be unclear whether the local ordering is actually causing the
pseudogap. Recent theoretical work has shown that proximity to the antiferromagnetic
insulator could independently create a pseudogap in the density of states, and an instability
towards forming ordered states. [83] The important physics of the pseudogap would then be
encapsulated in residual antiferromagnetism, the antiferromagnetic correlations not involved
in the formation of the ordered states.
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91
Chapter 5
The CuO2 Plane
5.1 Introduction
Although the materials that exhibit high temperature superconductivity vary widely in com-
position and structure, they are all layered compounds with one or more perovskite CuO2
plane per unit cell (Fig. 5.1a&b). As a result, the interesting highly-correlated electronic be-
havior of the HTS is thought to originate on these CuO2 planes. Their electronic structure
derives from Cu 3dx2−y2 and O 2px and 2py orbitals, with the anti-bonding configuration
of these orbitals sitting nearest the chemical potential (Fig. 5.1b). In its parent state,
one spin-half electron occupies every CuO2 plaquette, with strong electron-electron correla-
tions resulting in the formation of an antiferromagnetic insulator. Once some of the other
planes of the crystal structure remove electrons (create holes) on the CuO2 plane, the rest of
the strongly correlated behavior, including superconductivity and the pseudogap behavior,
emerges. It is thus of fundamental importance to make measurements that focus on the
properties of the CuO2 plane.
Both photoemission and scanning tunneling spectroscopy have been workhorses in diag-
nosing this correlated electronic behavior, in particular for their ability to measure the energy
dependence of the density of states. However, both techniques are surface-sensitive, and thus
92
A B
Figure 5.1: (A) This schematic drawing (side view) shows the positions ofthe metal ions in Bi2Sr2CaCu2O8+δ. The interesting many- body electronicbehaviour occurs on the pair of CuO2 planes (red) in every half-unit cell. Theother planes act as either dopant layers containing the extra oxygens (BiO,blue) or as spacer layers (Ca, green and SrO, yellow). (B) This is a schematicdrawing (top view) of the structure of the CuO2 plane common to all the HTSmaterials. Here, the red dots correspond to Cu atoms and the small greendots to O atoms. The unit cell of this plane, called a plaquette, is a squarecontaining four Cu atoms at the corners and four O atoms on the sides. Alsoshown are the Cu 3d orbitals and O 2p orbitals.
rely on having flat, clean surfaces on which to perform measurements. For this reason, these
techniques rely on mechanically cleaving samples before making measurements, a process
that breaks apart the two crystallographic plane in the crystal most weakly bonded to one
another. In general, nearly all the STM and photoemission studies on the HTS have been
performed on samples terminated by planes other than the CuO2 plane as a consequence of
this cleaving process. Most studies have focused on the compound Bi2Sr2CaCu2O8+δ, be-
cause it cleaves easily and exposes macroscopically flat regions for measurement. However,
in this case, cleaving exposes the BiO layer, [1] which sits 4.5A above the pair of CuO2
planes in the unit cell (Fig. 5.1a). This layer contains the excess oxygens used to dope
Bi2Sr2CaCu2O8+δ, [2, 3] and thus the BiO plane has been thought of as a doping plane,
93
D
A CB
Binding Energy (V)
Figure 5.2: (A) The tunneling conductance measured by STM on an under-doped Bi2Sr2CaCu2O8+δ sample is shown here as a function of temperature(adapted from Ref. [4]). (B) This is the spectrum measured by photoemis-sion near the (0, π) point of the Brillouin as a function of temperature on anunderdoped sample with TC = 83K. (adapted from Ref. [5]) (C) A phase dia-gram can be constructed from either measurement which resembles the phasediagram constructed from in-plane resistivity measurements (D). Figures (C)and (D) have been adapted from Ref. [6]
with the SrO and Ca planes acting as spacer layers.
The majority of the energy-resolved density of states information known about the
cuprates has thus been gathered using surface-sensitive techniques Bi2Sr2CaCu2O8+δ sam-
ples with BiO surface termination, and has been assumed to be representative of the prop-
erties of the sub-surface CuO2 planes. This assumption appears valid on a qualitative level
because of the rough level of agreement between the two surface-sensitive techniques, and
between them and bulk probes. The density of states measured by STM shows sharp peaks,
located at the edge of the energy gap, which appear only for samples in the superconduct-
ing state (Fig. 5.2a). [4] Similarly, the sharp peak seen in photoemission data near the
zone boundary, where the energy gap reaches its maximum value, appears only below TC
94
Tip
Sample
d
A B
Figure 5.3: (A) In an STM measurement, the tunneling current depends ex-ponentially on the distance d separating the sample and the tip. (B) In aphotoemission measurement, monochromatic light shined on a sample causesthe emission of a photoelectron. The penetration depth of light into the samplesurface is typically tens of Angstroms. However, the degree of surface sensi-tivity can be enhanced by the escape depth of the photoelectron- the depthinside the material from which a photoelectron can be emitted into vacuumwithout interacting with the material. (Figure adapted from [12])
(Fig. 5.2b). [5, 7] In addition, both techniques see a significant density of states at the
Fermi energy, and a pseudogap, for temperatures between TC and T ∗ in underdoped sam-
ples. [8, 9, 10, 4] The two experimental techniques, despite making measurements using
different quantum processes, have even been shown to be in agreement in the subtle details
of the spectra they measure. [11] Most importantly, the doping-temperature phase diagram
that can be constructed using these surface-sensitive techniques qualitatively agree with the
phase diagram constructured using techniques which probe CuO2 planes in the bulk (Fig.
5.2c&d). [6]
The process of making both measurements, however, is surface-sensitive to a disturbing
degree. In STM, tunneling current between the tip and the sample depends exponentially
on the distance separating them. Because the nearest CuO2 plane lies 4.5A beneath the
common BiO surface in Bi2Sr2CaCu2O8+δ, the matrix element suppresses direct tunnel-
ing into the CuO2 plane by four orders of magnitude in comparison to tunneling processes
involving the BiO plane. In photoemission, calculations based on experimental evidence
95
A B
Figure 5.4: (A) This is a three-dimensional rendition of an STM topographtaken by Ref. [15] on a single crystal of Bi2Sr2CaCu2O8+δ cleaved in air.Although spectra were taken on these terraces, the dirt present on the samplesurface makes it difficult to determine which layer of the crystal structure thesemeasurements were taken on, or whether the spectra are representative. (B)This figure shows an STM topograph of a single crystal of Bi2Sr2CaCu2O8+δ
cleaved in ultra-high vacuum rendered in a way to show the terraces of thismulti-stepped region. The investigators were unable to take spectra on thissample. Although they were able to determine the identity of the terraces,none of the exposed plains was a CuO2 plane (Figure adapted from Ref. [16]).
from the other crystallographic planes indicate that the escape depth for the emitted pho-
toelectrons is only ≈ 3A, meaning that the other planes must play an important role for
photoelectrons originating from the CuO2 planes (Fig. 5.3). [13, 14] Because the effect of
the other crystallographic layers on both measurements is unknown, performing these mea-
surements on samples terminated by the CuO2 plane is necessary to confirm that data taken
on BiO-terminated samples is representative of the behavior of the CuO2 plane in the bulk.
In this chapter, we will present the first conclusive topographic and electronic density of
states measurements on a single CuO2 plane at the surface of a HTS using STM. [17] There
have been some STM experiments performed on HTS samples terminated by different crys-
tallographic planes. [18, 15, 16, 19, 20] However, none of these has managed to conclusively
identify a CuO2 plane at the surface (Fig. 5.4). Such an identification requires a careful
correlation of STM topographs with crystallographic data, which, in turn, requires exper-
iments to be conducted entirely in an ultra-high vacuum environment to prevent surface
contamination. [19] Clean, reproducable atomically-resolved topographs are also required
96
to make such an identification, and to confirm the quality of the tunnel junction in making
spectroscopic measurements. [20] Our measurements indicate that the CuO2 plane can form
a stable termination layer on the surface of Bi2Sr2CaCu2O8+δ, with a lattice similar to that
of the plane in the bulk of the sample. The spectroscopic measurements made on this plane
are qualitatively different than those made on the common BiO surface, and, in comparison,
tunneling directly into the surface CuO2 plane appears strongly suppressed near the Fermi
energy. We suggest that the reason for this discrepancy is a change in doping for the surface
CuO2 plane resulting from the absence of the other crystallographic planes of the unit cell.
Although more work is required to determine the electronic phase of the surface CuO2 plane,
for which we consider all the phases of the HTS phase diagram to be a possibility, this work
opens up a new avenue to studying the physics of the high temperature superconductors.
5.2 Results
To search for a surface CuO2 plane, we return to the Bi2Sr2CaCu2O8+δ thin films used
in the first part of Chapter 3. Cleaving thin films requires significantly more force than
cleaving single crystals, most likely because the former contain a higher density of inter-
plane defects. This suggests that, unlike single crystals, the thin films might not cleave at the
weakest bond of the crystal structure (intercell BiO-BiO bond) for all regions of the surface.
The slightly underdoped 1000A thick films were grown using molecular beam epitaxy, and
demonstrated a resistive transition at TC = 84K. They were cleaved mechanically in ultra
high vacuum (UHV) at room temperature prior to making measurements using a home-
built UHV STM operating at T = 4.2K (LTSTM). The majority of the surface contains
wide terraces (100 − 1000A) that yield topographs (Fig. 5.5a) and conductance spectra
(Fig. 5.5b) reminiscent of those previously measured on BiO terminated single crystals.
[21, 22, 23, 24, 25, 4] In addition to these BiO terminated regions, we occasionally find
97
0
1
2
-200 -100 0 100 200
dI/d
V (
nS
)
V (mV)
A B
Figure 5.5: (A) This is an STM topograph of a 100A × 100A area taken ona BiO-terminated region of Bi2Sr2CaCu2O8+δ (I = 200pA and V = 200mVtunnel junction). (B) Although the conductance spectra measured on thisplane vary from location-to-location, they all feature an inchoate gap at theFermi energy (V = 0) flanked by sharp conductance peaks (dc junction impe-dence Rj = | − 200mV |/200pA = 1GΩ). This Figure has been adapted fromRef. [17].
smaller terraces on the sample that are terminated by other crystallographic layers.
We will focus on STM measurements taken near the step edge in Figure 5.6a, which we
find to have subunit cell height. Tunneling conductance measurements taken far from this
step edge on the top terrace (Fig. 5.6b) resemble those taken on the majority of the surface of
the thin film, and those measured on cleaved single crystals. We can thus conclude that the
BiO plane terminates the top terrace. Contrary to the expectation that the spectra will be
unaffected by the identity of the top layer of the sample, the spectra measured far away from
the step edge on the bottom terrace (Fig. 5.6b) are qualitatively different. To determine the
identity of the bottom terrace, we need to make an accurate determination of the height of
separation between the top (BiO) and bottom (unknown) terraces, and compare the result
to the known separation between the crystal planes of Bi2Sr2CaCu2O8+δ (Fig. 5.6c). [3]
Averaging all the scanlines in the topographic image of the step revealed the height to be
98
0
2
4
6
8
0 25 50 75 100Distance (Å)
Tip
Heig
ht
(Å)
BiO
SrO
Ca
CuO2
CuO2
topbottomA B
C
BiOCuO2
BiOCuO2
BiOCuO2
D
Figure 5.6: (A) This STM topograph of a 100A × 100A region of the sampleshows a sub-unit cell height step edge (I = 50pA and V = −200mV ). Thedata has been rendered in a way to bring out the atomic corrugation on boththe top and bottom terraces. (B) A line of tunneling conductance spectra wastaken perpendicular to the step edge, at the distances indicated. Both thetopmost and bottommost spectra (red curves) were taken at a distance about50A away from the step edge. (C) The linescans comprising the topographin (A) were averaged together, and compared with the distance of separationbetween the crystalligraphic planes of Bi2Sr2CaCu2O8+δ known from bulkscattering techniques. [3] (D) This is a schematic diagram of the sub-unit cellheight step edge in (A). Parts (A)-(C) of this figure have been adapted fromRef. [17].
99
A B
0.01
0.1
1
-200 -100 0 100 200
Noise
Floor
0
1
2
-200 -100 0 100 200
BiO
CuO2
dI/dV(nS)
V(mV)
C
V(mV)
dI/dV(nS)
Figure 5.7: (A) This is an STM topograph of a 64A × 64A area taken on aCuO2-terminated region of Bi2Sr2CaCu2O8+δ (I = 200pA and V = 200mVtunnel junction). The inset shows a 8A × 8A topograph of a subset ofthis region (adapted from Ref. [17]). (B) This is a set of typical tunnel-ing conductance spectra, taken on the region in (A) (dc junction impedenceRj = | − 200mV |/200pA = 1GΩ). For comparison, we also show a representa-tive spectrum from a BiO-terminated surface taken under the same conditions.(C) This is the same plot as in (B), except on a log-lin scale. The noise floorof our conductance measurement is also shown.
≈ 8A. Sources of error, arising from the miscalibration of our instrument or a difference in
electronic structure between the two planes leading to enhanced tunneling current, were all
estimated to be much smaller (0.1 − 0.2A) than the separation between any two adjacent
layers in the crystal structure (≈ 2A). Consequently, the correlation between the height
of this step edge and the known separation between the crystallographic layers (Fig. 5.6c)
identifies a single CuO2 plane as terminating the lower terrace. We have used this procedure
to identify several CuO2 terraces at the surface of different Bi2Sr2CaCu2O8+δ samples grown
under the same conditions. Because of the large resistivity anisotropy in Bi2Sr2CaCu2O8+δ
(ρc/ρab ≈ 100), the coupling between CuO2 planes in adjacent half-unit cells is expected to be
very weak in this material. Consequently, we believe our measurements to be representative
of a single CuO2 plane doped only by the crystal planes beneath it.
Having successfully identified a CuO2-terminated region of the sample, we now turn
to measuring its unperturbed electronic and structural properties far from the step edge.
100
Topographic images of this surface, shown in Figure 5.7, show a well-ordered square lattice.
The b-axis superlattice distortion is seen to modulate this atomic structure at 45 to the
Cu − Cu nearest neighbor direction, in agreement with other measurements on bulk CuO2
planes. [3, 26] Spectroscopic measurements made on the CuO2 surface show a density of
states that is qualitatively different than those measured on BiO surfaces in four ways (Fig.
5.7b): the CuO2 spectrum has a wider energy gap, weaker conductance peaks at the edge
of this gap, different high energy features, and, most importantly, a strongly suppressed
tunneling conductance near the Fermi energy. Tunneling into the surface CuO2 plane is so
strongly suppressed near the Fermi energy that the conductance is beneath the noise floor
of our equipment (Fig. 5.7b). Conductance measurements taken over distances between
the atomic corrugation spacing and several hundred Angstroms show qualitatively similar
features, indicating the features in the spectra highlighted above are representative of those
measured on a surface CuO2 plane. The contrast between the density of states measured
on BiO-terminated and CuO2-terminated regions of Bi2Sr2CaCu2O8+δ is the main result
reported in this chapter, and demonstrates the importance of the identity of the surface layer
in STM measurements on this material.
5.3 Interpretation
Although the effect of the other crystallographic planes on STM measurements is still un-
known, we believe that tunneling conductance measurements made on BiO-terminated sam-
ples are representative of the electronic behavior of the subsurface CuO2 planes. As outlined
previously, basic features in the spectra measured by both STM and photoemission agree
with one another, despite the fact that the two techniques probe materials in fundamentally
different ways. These features can be used to construct a phase diagram that agrees with
probes that measure the properties of CuO2 planes in the bulk. In addition, a comparison
101
of the Fermi surface measured by photoemission [12, 27] and electronic structure calcula-
tions [28] has indicated that only the CuO2 planes in Bi2Sr2CaCu2O8+δ have states near
the Fermi energy. This suggests that the low-energy features in the tunneling conductance
measured with STM on the BiO surface should agree with the expectation that d-wave
BSC-like superconductivity is present on the sub-surface CuO2 planes. To test this idea, we
will model the tunneling conductance as
dI
dV(V ) ∝
∫
d2~k∫
dUn(~k, EF + V )|M(~k)|2e−(EF +V −U)/2σ2
, (5.1)
where n(~k, EF + V ) = n0V/√
V 2 − ∆(~k)2 is the BCS density of states, n0 is the den-
sity of electronic states at the Fermi surface in the absence of superconductivity, ∆(~k) =
∆0[cos kxa − cos kya] is the energy gap for a d-wave superconductor, and σ is a broadening
factor. Assuming that the tunneling matrix element is isotropic, the calculated tunneling
conductance agrees with the STM data measured on the BiO plane at energies near the
Fermi energy. The value of the energy gap, ∆0 = 40mV , is similar to what we would expect
for a nearly optimally-doped sample, and the value for thermal broadening, σ = 6mV , is
consistent with the conditions of our measurement. More detailed models, derived using the
band dispersion measured by photoemission and interaction with the fluctuating antiferro-
magnetism measured by neutron scattering (see Chapter 3), have been shown to reproduce
the precise features of the STM tunneling spectrum in the superconducting state out to
V = ±200mV . [11] Although no definitive experiment has been done to identify the ef-
fect of the other planes, a preponderance of evidence indicates that STM measurements on
BiO terminated Bi2Sr2CaCu2O8+δ samples do probe the properties of the subsurface CuO2
planes.
In contrast to tunneling on the BiO surface, the process of tunneling directly into a
CuO2 plane at the surface is not complicated by the presence of the other crystallographic
planes. However, the tunneling conductance measured on the CuO2 terminated surface
differs significantly from the conductance measured on the BiO terminated surface, particu-
102
-200 0 200
1
2
dI/
dV
(n
S)
V (mV)
-200 2000
V(mV)
dI/d
V (
arb
. u
nit
s)
1
2
Figure 5.8: (A) For in-gap energies, the qualitative features of a tunnelingconductance spectrum measured on the BiO surface (left) are qualitativelyreproduced using the model discussed in the text (right).
larly for in-gap energies, where tunneling into the CuO2 surface is dramatically suppressed.
Moreover, as we have just shown, the spectrum on the BiO surface qualitatively matches
our expectations that the electronic state of the sample is described as a BCS-like d-wave
superconductor. Thus we must conclude that the process of exposing the CuO2 plane at
the surface dramatically modifies its electronic character compared to when it is beneath
a BiO terminated surface, or when it is in the bulk of the sample. Such a situation could
arise if exposing the plane at the surface leads to a structural modification of the CuO2
lattice. To address this possibility, we can compare the topographic images obtained on
both the BiO and CuO2 surfaces (Fig. 5.5a and Fig. 5.7a). The atomic corrugations seen
in topographs on BiO terminated samples are believed to correspond to Bi atoms, which
sit above the Cu atoms of the CuO2 plane, and those on the CuO2 plane are believed to the
the Cu atoms themselves. A comparison of both topographs reveals the nearest neighbor
corrugation distance to be equal (3.8A), and to be consistent with the Cu − Cu distance
measured in the bulk. [3] Furthermore, as mentioned previously, the b-axis corrugation is
seen to have the same periodicity for both the surface and bulk CuO2 planes. Although the
103
error in determining both these distances using STM topographs of the surface CuO2 plane
is large (≈ 0.1A) compared to the error in determining them using scattering techniques
on planes in the bulk (typically an order of magnitude better), this agreement leads us to
believe that the CuO2 lattice is not distorted at the surface.
We believe the most likely complication introduced by exposing the CuO2 plane at the
surface is precisely the absence of the other crystallographic planes. In order to expose the
CuO2 plane, we have had to remove many of the crystallographic layers of Bi2Sr2CaCu2O8+δ
on one side of this plane (Fig. 5.6d). Chief amongst these is the second CuO2 plane in each
half unit cell. However, STM conductance spectra from the BiO surface of Bi2Sr2CuO6+δ,
which has a single CuO2 plane per half unit cell, have a similar shape for in-gap energies
as the spectra taken here on the BiO surface of Bi2Sr2CaCu2O8+δ, suggesting that the
removal of the other layers might play a more significant role. [29] Indeed, the other missing
planes- one Ca layer, one SrO layer, and one BiO layer- are crucial for the charge balance
in this compound. Doing simple valence charge counting, the oxygen atoms attract two
electrons apiece, the strontium, copper and calcium atoms give up two electrons apiece, and
the bismuth atoms give up three electrons apiece. Consequently, a single CuO2 plane wants
to attract two electrons to become chemically stable. In a complete half unit cell, the two
SrO layers are charge neutral, the two BiO layers each donate an electron to each of the
two CuO2 planes, and the one Ca layer donates an electron to each of the two CuO2 planes,
resulting in chemically stable CuO2 planes. In the geometry that exposes a single CuO2
plane at the surface, we have a BiO, SrO, and a CuO2 plane, meaning that the CuO2 plane
is not chemically stable because it is missing one electron per plaquette. Because topographic
data of this plane indicate it is both stable and structurally similar to a bulk CuO2 plane,
some complex charge redistribution is most likely happening in the vicinity of this plane.
Therefore, we must consider the possibility that we have dramatically changed the electronic
phase of this plane.
104
The most obvious possibility for the electronic phase of the surface CuO2 plane is an insu-
lator, with the observed energy gap being associated with injecting or removing electrons into
the system. Our data, however, place some stringent requirements on a possible insulating
phase. The most stringent of these requirements is that the insulator must be particle-hole
symmetric, as the chemical potential is located in the center of the 2∆ = 120mV energy
gap seen in the conductance spectrum. In a simple (band) insulator, the chemical potential
is pinned by surface defects. Because we have seen the same electron-hole symmetric en-
ergy gap on several surface CuO2 planes, with different terrace sizes and defects, we believe
that the surface CuO2 plane is not a simple insulator. Another promising explanation is
that the surface CuO2 plane is itself conducting, but the observed spectrum results from a
Coulomb blockade, as seen in disordered or granular systems with STM. [30] These systems
are typically modeled as a small, isolated conductive element in between an electron source
and an electron drain. In a simple model, the small conductive element must be capacitively
charged, at an energy cost of Ec = e2/2C before it conducts (here, e is the charge of an
electron and C is the capacitance between the conductive element and a source of electrons).
Moreover, this energy cost can be roughly the same for electron injection or removal, cor-
responding to positive and negative energies. Modeling the tip-sample tunneling junction
as a parallel-plate capacitor with a separation of d = 6A, we find that the observed energy
gap corresponds to a grain with diameter 100A. However, we measure the same energy
gap over distances much larger than 100A, without encountering any grain boundaries. In
sum, the requirements that the spectra both contain a particle-hole symmetric energy gap
and remain roughly spatially homogeneous suggest that some strongly correlated electronic
behavior exists on the surface CuO2 plane.
The effect of removing the layers on top of the CuO2 plane could be simply a change in
doping level for the surface CuO2 plane compared to the plane in the bulk. This is supported
by the fact that the lattice structure of this plane appears undisturbed by the process of
105
exposing it at the surface. However, although our samples were seen to have a resistive
superconducting transition at T = 84K both before and after cleaving, the removal of the
planes could change the local doping level quite dramatically. As a result, we consider the
possibility that any of the phases of the HTS phase diagram might be present on this plane.
One possibility is that the doping level of the surface CuO2 plane has not changed
much, and it is still superconducting. Assuming that the superconducting state is similar
to that of bulk CuO2 planes, we expect the density of states at low energies to have a
BCS form with a d-wave energy gap. As we demonstrated earlier, if we assume that the
STM tunneling matrix element is isotropic, this calculation yields a tunneling conductance
with a far greater amplitude inside the energy gap than what is measured on the surface
CuO2 plane (Fig. 5.7b and Fig. 5.8). However, theoretical efforts have proposed that the
mechanism of tunneling an electron into the CuO2 plane along the c-axis, as in the STM
geometry, results in anisotropic tunneling. [31, 32] These theoretical works have argued that
tunneling along the c axis direction does not occur directly into Cu 3d states, which lie flat
in the CuO2 plane (Fig. 5.9a). Instead, tunneling is facilitated by the orbital that extends
the furthest out of the plane- the 4s orbital of the Cu atoms. In the STM geometry, this
means that tunneling probes the electronic states near the Fermi energy in the CuO2 plane
through a two step process, where electrons at the Fermi energy in the tip first tunnel into
the Cu 4s orbital, which lies above the Fermi energy, and then from this orbital into the Cu
3d orbital, which lies near the Fermi energy. However, the coupling of electrons between the
Cu 4s and Cu 3d orbitals introduces an anisotropic matrix element that has a d-wave form
|M(~k)|2 ∝ [cos kxa−cos kya]. Simply, the tunneling of electrons into the CuO2 planes occurs
preferentially along the directions where the energy gap is at its maximum. We show the
effect of incorporating this matrix element into the calculation of the tunneling conductance
using a BCS density of states and a d-wave order parameter in Figure 5.9b.
Accounting for the anisotropy of tunneling into the CuO2 plane arising from the symme-
106
Cu
Cu
Cu
Tip
Cu 4S
Cu 3dSide
View
A B
Figure 5.9: (A) Shown is a schematic model for tunneling into the CuO2 plane,as discussed in the text. (B) This model of tunneling results in the anisotropictunneling of electrons into the CuO2 plane. Shown here is a conductancecurve produced by the model of tunneling conductance discussed in the textoverlayed on an STM conductance spectrum taken on the CuO2 plane (Figureadapted from Ref. [17]).
try of the molecular orbitals involved in the tunneling process gives a fairly close match to
the measured low-energy conductance. The best fit with the experimental data is obtained
using ∆0 = 60mV , which is larger than the value expected for samples with a TC of 84K.
However, this value of the energy gap is close to that previously reported on underdoped,
but still superconducting, samples (δ = 0.1). [6] Furthermore, it has been suggested that
underdoping the sample will cause the broad hump seen at negative energies on BiO termi-
nated samples to shift to larger negative energies, providing a potential explanation for the
feature seen at higher negative energies on the surface CuO2 plane. [11] However, to obtain
the best match between the calculation and the data, we have had to use an unphysically
large amount of energy broadening (σ = 12mV ). In addition, the feature seen at higher
positive energies on the CuO2 plane has no analog in the BiO-terminated data. However,
both these can be rationalized as subtle differences caused by the removal of the other crys-
tal planes. More troubling is the fact that the same physical arguments used for invoking
this anistropic tunneling matrix element can be made for tunneling over BiO-terminated
107
samples. The constrained c-axis tunneling that results from this model was, in fact, first
proposed as an explanation for the large anisotropy between the in-plane and out-of-plane
resistivity in the HTS. [31, 32] Although we cannot rule out the possibility that the surface
CuO2 plane is superconducting, it seems unlikely that anisotropic tunneling arising from the
symmetry of the atomic orbitals occurs only when the CuO2 plane is at the surface, and not
when it is in the bulk.
Another possibility is that the surface CuO2 plane is in the low-temperature pseudogap
regime that exists for dopings in between the superconductor and antiferromagnetic insulator.
This interpretation is tempting because the physics of the pseudogap regime is unsettled,
many competing proposals have been made (Chapter 4), and this would provide a new, direct
way of examining a CuO2 plane in the pseudogap regime at low temperatures. However,
in the cases where these theoretical proposals predict the density of states, they predict a
pseudogap- an energy gap with a finite number of states at the Fermi energy. In contrast, we
find that the tunneling conductance is strongly suppressed at the Fermi energy, and drops
below the noise floor for our measurement within 10mV of the Fermi energy. Moreover,
recent STM measurements on lightly doped Ca2−xNaxCu2O4Cl2 see a density of states with
remarkably different characteristics than the spectra measured by us on the surface CuO2
plane. Notably, the spectra measured on Ca2−xNaxCu2O4Cl2 have a ’real’ pseudogap, with
a measurable number of in-gap states (Fig. 5.10). [33] Nevertheless, the doped CuO2 plane
might possess an instability to different kinds of ordered states, depending on factors other
than doping and temperature. The electronic state of the surface CuO2 plane might thus
still be related to one of the various exotic states proposed for the pseudogap regime.
The final possibility is that the surface CuO2 plane is in the lightly-doped antiferromag-
netic insulator regime of the HTS phase diagram. At dopings below where superconductivity
occurs, resistivity data shows a crossover from metallic behavior at high temperatures to in-
sulating behavior at low temperatures. The authors of this work suggest that the reason
108
Figure 5.10: STM conductance spectra taken on the compoundCa2−xNaxCu2O4Cl2 for a doping of x = 0.06 (insulating state), x = 0.08(zero temperature pseudogap regime), and x = 0.1 (superconducting state) allshow an ill-defined energy gap reminiscent of a pseudogap. (Figure adaptedfrom Ref. [33]).
is that charge carriers exist at the chemical potential at these dopings, but are localized.
[34] They further note that the hole mobility scales with doping in a fashion similar to the
average spacing between holes (3.8A/√
δ). This could be consistent with our observation
of a spatially homogeneous energy gap if we treat the surface CuO2 plane as a collection
of localized holes. The energy gap in the spectra on this plane would then be the result
of Coulomb blockade associated with tunneling into an island the size of this localization
length scale. Moreover, an island of the same size, having the same energy gap, would sim-
ply follow the tip wherever we attempted to tunnel into the sample. We can attempt to
determine whether the size of the observed energy gap is consistent with this scenario. The
observed energy gap of Ec = 60mV indicates an island size of 100A from using the Coulomb
gap equation. The doping level of the sample could then be calculated using the average
separation between holes at a particular doping. Our energy gap translates into a doping
level of δ = .01, where the sample is expected to show this localization behavior. Although
we cannot make a conclusive claim that the surface CuO2 plane is in the doped insulating
regime, this explanation provides a self-consistent explanation of the data.
109
Clearly, more data is needed before we can distinguish which of the electronic phases
is present on the surface CuO2 plane. If step edges could be found cutting through this
plane along the 110 direction, we could test for the presence of Andreev bound states,
which should only be present in a d-wave superconductor (Chapter 1). If large data sets over
large regions of the sample had been taken, as in Chapter 3, we could examine the spatial
structure of the density of states at a given energy to look for the presence of local charge
ordering. Finally, if we had varied the tip-sample separation by a significant distance (≈ 1A),
we would have been able to test whether the observed energy gap came from a Coulomb
blockade effect arising from the localization of electrons in the doped insulator. Changing
the tip-sample separation would have changed the capacitance, and altered the size of the
Coulomb charge gap.
Irrespective of what the electronic phase of the surface CuO2 plane is, our demonstration
that it is possible to tunnel directly into a CuO2 plane opens a new way to probe the
HTS. Because its crystal structure appears unaltered by exposing it at the surface, STM
measurements made on the surface CuO2 plane should be representative of the behavior of
perovskite CuO2 planes in general. Furthermore, the primary effect of removing the other
crystal layers appears to be a change in the carrier concentration of the surface CuO2 plane.
As a result, it should be possible to measure properties of the HTS, which are thought
to be nothing more than doped CuO2 planes, by making STM measurements directly on
CuO2-terminated samples, with the added benefit that the process of measurement is not
complicated by the presence of the other crystallographic planes.
5.4 References
[1] P. Lindberg et al., Phys. Rev. B 39, 2890 (1989).
[2] J. Tarascon et al., Phys. Rev. B 37, 9382 (1988).
110
[3] H. Heinrich, G. Kostorz, B. Heeb, and L. Gauckler, Physica C 224, 133 (1994).
[4] C. Renner et al., Phys. Rev. Lett. 80, 149 (1998).
[5] D. Feng et al., Science 289, 277 (2000).
[6] J. Tallon and J. Loram, Physica C 349, 53 (2001).
[7] H. Ding et al., Phys. Rev. Lett. 87, 227001 (2001).
[8] M. Norman et al., Nature 392, 157 (1998).
[9] A. Loeser et al., Science 273, 325 (1996).
[10] H. Ding et al., Nature 382, 51 (1996).
[11] B. Hoogenboom et al., Phys. Rev. B 67, 224502 (2003).
[12] A. Damascelli, Z. Hussain, and Z.-X. Shen, Rev. Mod. Phys. 75, 473 (2003).
[13] M. Norman, M. Randeria, H. Ding, and J. Campuzano, Phys. Rev. B 59, 11191 (1999).
[14] A. Zakharov, I. Lindau, and R. Yoshizaki, Physica C 398, 49 (2003).
[15] H. Murakami and R. Aoki, J. Phys. Soc. Jpn. 64, 1287 (1995).
[16] S. Pan, E. Hudson, J. Ma, and J. Davis, Appl. Phys. Lett. 73, 58 (1998).
[17] S. Misra et al., Phys. Rev. Lett. 89, 87002 (2002).
[18] T. Hasegawa and K. Kitazawa, Jpn. J. Appl. Phys. 29, 434 (1990).
[19] K. Kitazawa, Science 271, 313 (1996).
[20] S. Sugota, T. Watanabe, and A. Matsuda, Phys. Rev. B 62, 8715 (2000).
[21] A. Yazdani et al., Phys. Rev. Lett. 83, 176 (1999).
111
[22] E. Hudson et al., Science 285, 88 (1999).
[23] S. Pan et al., Nature 403, 746 (2000).
[24] E. Hudson et al., Physica B 329, 1365 (2003).
[25] E. Hudson et al., Nature 411, 920 (2001).
[26] M. Asensio et al., Phy. Rev. B 67, 14519 (2003).
[27] J. Campuzano, M. Norman, and M. Randeria, in Nobel Lectures, Physics 1981-
1990, edited by K. Bennemann and J. Ketterson (Springer-Verlag, ADDRESS, 2004),
Chap. Photoemission in the High-TC Superconductors.
[28] D. Singh and W. Pickett, Phys. Rev. B 51, 3128 (1995).
[29] M. Kugler et al., Phys. Rev. Lett. 86, 4911 (2001).
[30] A. Hanna and M. Tinkham, Phys. Rev. B 44, 5919 (1991).
[31] S. Chakravarty, S. Sudbo, P. Anderson, and S. Strong, Science 261, 337 (1993).
[32] O. Anderson, A. Lichtenstein, O. Jepson, and F. Paulson, J. Phys. Chem. Solids 56,
1573 (1995).
[33] Y. Kohsaka et al., Phys. Rev. Lett. 93, 97004 (2004).
[34] Y. Ando et al., Phys. Rev. Lett. 87, 17001 (2001).
112
Chapter 6
Conclusion
We began this thesis motivated by the goal that scanning tunneling microscopy can teach
us new information about the high temperature superconductors. The characteristic length
scales of the HTS phase diagram are all on the atomic length scale- the coherence length for
superconductivity is ≈ 16A in-plane, and the various ordered states discussed in relation to
the HTS phase diagram typically have length scales on the order of tens of Angstroms. STM
is capable of measuring the density of states on atomic length scales, and correlating these
measurements with topographic data of the surface it takes the spectroscopic measurements
on. Through the course of this thesis, we have demonstrated how the nanoscale view of the
HTS provided by STM has lead to new discoveries throughout the phase diagram.
In the superconducting state, we investigated the spectacular changes that occur to
the parent d-wave superconducting state upon the introduction of certain types of defects.
Although the existence of the novel zero-energy state found at 110 step edges could be
predicted using the existing theory, we confirmed that it does form a one-dimensional state
at the interface. More importantly, the absence of this state at 110 twin boundaries,
where the established theory predicted it should also occur, opens up the issue of how
twin boundaries affect the superconducting state. As it is exceptionally difficult to grow
untwinned samples, and the effect of twin boundaries on other experiments is still debated,
113
a definitive answer to this issue is needed. Although our data lend preliminary support to
the idea that 110 twin boundaries nucleate a subdominant s-wave superconducting order,
in itself an exciting possibility, more data will be needed to make a definitive statement. The
most promising line of inquiry would be making measurements on 100 oriented defects,
which are not expected to give rise to either of these novel effects.
Far away from these extended structural defects, we found that the in-gap electronic
states form modulated patterns in the density of states at T = 40K. Although the obser-
vation of modulated patterns over seemingly pristine HTS samples was not a new discovery
in itself, as they had been seen before at low temperatures, it represented a landmark step
forward for STM. This was the first time detailed spectral mapping had been accomplished
by STM on a correlated electronic system at elevated temperatures. That we were able
to make meaningful measurements, which took days to acquire, with the potentially fatal
complication that the tunnel junction becomes unstable due to thermal fluctuations, opens
up the possibility for exploring the temperature dependence of the density of states with
atomic precision in any number of materials systems for the first time. The analysis of the
modulations in terms of perturbative corrections to the density of states had also been in-
vestigated thoroughly in the literature. We demonstrated that the scattering interference
scenario qualitatively described the changes seen in these patterns between 4K and 40K,
including the disappearance of 5 of the 6 modes seen in the low temperature data due to
thermal fluctuations. Perhaps even more importantly, it lent credence to the idea that the
formalism developed in doing these calculations can be used to diagnose, if not describe, any
modulated pattern seen by STM.
In contrast to the superconducting state, we lack even a rough phenomenological under-
standing of the physics underlying the pseudogap regime. For this reason alone, our discovery
that the in-gap electronic states of the pseudogap regime form disperionless bond-oriented
modulations is perhaps the most significant work presented in this thesis. Having understood
114
the presence of the patterns in the superconducting state to be a consequence of scattering
interference, which nominally requires the existence of quasiparticles, it came as a shock that
these patterns were present above TC , where quasiparticles have been thought to not exist
at all. The scattering interference formalism developed in attempting to understand the
superconducting state data proved to be invaluable in diagnosing the possible origin of these
modulations. We had hoped to be able to identify which of the candidate electronic Green
functions proposed for the pseudogap state were consistent with the observed modulations.
The failure of any Green function to describe the dispersionless modulations led us to the
conclusion that the problem lay in the scattering interference model itself. Moreover, the
reasons for this failure suggested that these patterns represent the signature of some locally
ordered state whose appearance correlates with the pseudogap energy scale.
Future works will have to determine whether the observed local order is the signature of a
thermodynamically significant phase. Our measurements lead to the tantalizing possibility
that this local order causes the pseudogap in the density of states (or vice versa), and,
coupled with similar results in other cases where superconductivity is suppressed, that this
order competes with superconductivity. However, we have merely determined that a single
point on the phase diagram shows this local order in one HTS compound. We still do
not know whether the appearance of the local order correlates with the appearance of the
pseudogap in the density of states throughout the phase diagram. Even if the local ordering
does not represent a thermodynamically significant phase, they could still be important to
the study of the HTS by explaining anomalies measured by other techniques. Ultimately,
the importance of this local ordering to the study of HTS lies in determining under what
conditions these patterns are seen, and what anomalies in other techniques their properties
are consistent with.
In the last experiment presented in this thesis, we measured, for the first time, the density
of states of a single CuO2 plane at the surface of a high temperature superconductor. The
115
highly correlated electronic behavior of the cuprates is thought to originate on these planes,
which are common to all the HTS materials. Although density of states measurements are
typically made using surface sensitive techniques on samples terminated by planes other than
the CuO2 plane, measurements made on the BiO surface of Bi2Sr2CaCu2O8+δ do appear
to be representative of the subsurface CuO2 planes. We thus would expect to measure
the same density of states when tunneling directly into a CuO2 plane at the surface. We
found, however, that the tunneling conductance is remarkably different in this situation.
This indicates that the process of removing the other planes, which we had thought would
simplify the process of measuring the properties of the HTS using STM, instead alters their
electronic state. However, the surface CuO2 plane has an identical crystal structure to the
structure of the plane in the bulk of the material, suggesting that the process of exposing the
CuO2 plane does nothing more than change its doping level. Although more experiments are
required to determine where on the cuprate phase diagram the surface CuO2 plane is, this
geometry might still be a new way to probe the HTS, which are nothing more than doped
CuO2 planes, without having to deal with the added complication of tunneling through the
other crystallographic layers in these compounds.
Through the course of this thesis, we have illustrated various techniques that will prove
valuable to the study of strongly correlated electron systems with STM in the future. In
general, observing how a material responds to a perturbation by measuring the density of
states near specific impurities, and comparing these results, can reveal information about the
correlations present in a material (Chapter 3). Determining whether there is a modulated
spatial structure to the density of states at fixed energies, and seeing how these modula-
tions evolve as a function of energy can help determine the electronic state of the sample
(Chapters 3 and 4). Finally, the ability to carefully correlate density of states information
with topographic data can reveal experimental situations that are completely inaccessible to
other techniques (Chapter 5). Most importantly, we have demonstrated that STM can be
116
used to map the temperature dependence of the density of states with unprecedented spatial
resolution. With this arsenal, STM appears poised to make a glut of important contributions
to the study of any number of correlated electron systems with short characteristic length
scales.
117
Appendix A
STM Methods
A.1 Construction
The basic requirements to make the tunneling measurements discussed in this thesis are that
the surface be atomically flat and clean, and that a tunnel junction can be established and
stably maintained for days at a time. Accordingly, the bulk of any STM system consists
of subsystems designed to prepare a sample surface for measurement, then bring a sharp
metallic wire (the tip) within a few Angstroms of the surface, and finally maintain the tip-
sample junction under unchanging conditions for several days. The two home built STMs
used to acquire the data presented in this thesis functionally have the same subsystems:
instruments for preparing clean metal surfaces, vacuum pumps to keep their surfaces clean,
a piezoelectric scan head that brings the tip close to the sample surface, cryostats to main-
tain a stable temperature, and spring-based vibration isolation to maintain the mechanical
stability of the tunnel junction. In this section, I will review how each of the two home-built
microscopes used in this thesis- one, a low temperature STM (LTSTM) and the other, a
variable temperature STM (VTSTM)- deals with each of these issues. The precise construc-
tion details of these instruments appear in the theses of Dan Hornbaker (LTSTM) [1] and
Michael Vershinin (VTSTM) [2], and will not be repeated here.
118
Both STMs share similar subsystems for sample preperation. Two types of samples are
routinely examined with these machines- noble metals (such as Au(111)), and metal oxides
(such as HTS). To clean the noble metal surfaces, the samples are first heated, for the
purpose of making the second step, sputtering, more effective. The LTSTM accomplishes
this through the resistive heating of a ceramic heating element located behind the sample
inside the sample holder. The VTSTM, instead, relies placing the sample inside an e-beam
heater, in which current is passed through a filament and then the hot electrons accelerated
at the back of the sample holder by means of an electric field. After heating, the sample
surface is ballistically sputtered clean. Ar gas is leaked into the chamber and ionized by
passing current through a filament. An applied voltage between the filament and the sample
surface accelerates the ionized Ar atoms at the sample. They collide with the atoms at
the surface, and ballistically knock them off, thus cleaning the surface. Finally, the now-
disordered surface is heated again to make the surface layer slightly molten. This causes
the atoms to rearrange themselves, and, upon quenching the sample to the temperature at
which measurements will be made, produces an atomically flat surface. Metal oxide surfaces
cannot be prepared by sputtering and annealing because most are doped with atoms, such
as Oxygen, that will leave the sample upon heating. Instead, a metal post is glued on to
the top of these samples and knocked off in vacuum, literally cleaving the samples in half.
This exposes a surface that is clean, having previously been confined to the interior of a
solid. In layered materials with weak interplane coupling, this also produces atomically flat
surfaces. Finally, both systems have turbomolecular pumps and ion pumps to maintain
background pressures of ≈ 5 × 10−10torr to prevent recontamination of these clean surfaces
while measurements are being made.
Once the sample surface has been prepared, the STM tip must be brought from an
essentially arbitrary macroscopic distance away from the sample to within a few Angstroms
of the surface in a controlled fashion. Both the microscopes employ what is referred to as
119
A B
Figure A.1: (A) A Besocke-style STM heads was used in both the LTSTMand VTSTM designs. The three tubes at the perimeter are the so-called legpiezos, and the one in the center is the so-called scan piezo (from Ref. [2]).(B) The sample holder consists of three ramps which sit on top of each of thethree leg piezos, and a center part where the sample sits.
a Besocke-style scan head to accomplish this (Fig. A.1a). [3] The scan head consists of
four piezoelectric tubes, three placed at the vertices of an equilateral triangle, and one in
the center. All the tubes have been metalized on their inside, and in four quadrants on
their outside. Application of a differential voltage across two opposing quadrants produces
a deflection of the tube along that axis, and the application of a voltage between the outer
electrodes and the inner one produces an elongation of the tube. The central tube (scan
piezo) carries the STM tip, and can be used to laterally scan the tip, and change the height
of the tip in response to the feedback loop. The outer tubes (leg piezos), on the other
hand, both carry the sample holder, and, through the electrically isolated balls at their ends,
provide the sample bias. The sample holder consists of a disk whose outer part consists of
three sloped ramps, underneath which sit the leg piezos, and whose inner part contains the
sample (Fig. A.1b-d).
To bring the tip to a distance where electrons can tunnel between the tip and the sample,
120
µN
kdd
Figure A.2: The procedue used to approach the sample operates like an inversepizza-toss. Shown are one of the three piezo legs and the sample holder/ rampassembly.
the legs are used to walk down the three ramps that sit on top of them (Fig. A.2). Walking
the sample towards the (comparably stationary) tip can be accomplished by suddenly con-
tracting the legs to disengage them from the surface of the ramp. Before the sample holder
falls back onto the legs, they are quickly deflected sideways. When the sample holder once
again makes contact with the legs, the contact points are located further ’down’ the ramps.
The voltage applied to the legs is then slowly relaxed to zero to restraighten the legs. In
this way, it is possible to make single ≈ 50A steps towards the sample. Finally, the feedback
loop is engaged, and the scanner extends the tip through a range of ≈ 200A. If, at any point
in this extention, tunneling current starts to flow, the feedback loop stabilizes the tip a few
Angstroms above the sample surface, and the approach is complete. If the tip extends to
the maximum of the scan piezo’s range, the feedback loop is opened, the tip retracted, and
another step is taken.
Having moved the sample to a position where we can find the surface by extending the
scan piezo, the STM must finally ensure that the tunnel junction remains stable. Slight
changes in position can have catastrophic effects, as the tip is merely 6A away from the sur-
face, and tunneling current depends exponentially on the tip-sample separation. This effect
is small for topographic measurements, as the feedback loop can compensate for changes in
121
tip-sample separation, but can make spectroscopic measurements impossible, as the feed-
back loop must be opened to take spectra. First, the temperature of the microscope stage
and sample must be kept constant to within 10mK, as the assymetric thermal expansion
of the scan piezo/ tip assembly compared to the leg piezo/ sample assembly can lead to
significant changes in tip-sample separation. Second, the microscope stage cannot vibrate.
To aid in vibration isolation, the entire experiment is set on top of an optical table inside of
an acoustic room. The optical table prevents sudden shocks in the floor, such as dropping
a hammer, from propogating into the system, and crashing the tip into the sample. The
acoustic room stops ambient room noise from vibrating the tip-sample assembly (STMs with
poor acoustic shielding can make excellent low-bandwidth microphones). However, the bulk
of the low-frequency vibration isolation, and the thermal components, have been treated
differently in the two microscopes used here.
The way the LTSTM and VTSTM solve these problems have dramatically influenced
how they were constructed. In the LTSTM, the microscope assembly is at the end of a
large pendulum which sits in the sealed cavity of a liquid Helium dewar (Fig. A.3). [4]
An exchange gas intruduced into this cabity helps maintains the microscope at a constant
T ≈ 4.5K. Because of the large thermal mass that remains cold, the microscope assembly
doesn’t warm up even when refilling the dewar. This means that the LTSTM is capable of
examining a single sample for months at a time. The entire pendulum assembly is attached to
the bottom of the vacuum chamber under which it sits using vacuum bellows. This acts like a
spring, damping vibrations from traveling from the vacuum chamber into the pendulum. In
addition, the pendulum is suspended using a set of three low-vacuum bellows whose internal
volume is connected to a small gas (or vacuum) reservoir. The pressure of the reservoir
can be changed, raising and lowering the pendulum. In addition, these reservoirs act like
dashpots for the bellows, which act like springs. Combined, they provide excellent low-
frequency vibration isolation, with tunable dissipation, for the microscope stage. However,
122
Optical
Table
Helium
Dewar
Isolation
Bellows
Vacuum
Pumps
Pendulum
Microscope
Exchange
Gas
Load Lock/
Cleaving Chamber Surface Prep Chamber
Sample
Manipulation
A B
Figure A.3: (A) This schematic diagram of the LTSTM highlights the key partsdealing with surace preperation, vibration isolation, sample manipulation, andtemperature regulation (adapted from Ref. [1]) (B) The microscope stage ofthe LTSTM sits on a linear motion stage which lifts the ramp and the sampleholder, which plugs into the ramp (from Ref. [1]).
due to the physical size of this setup, large manipulators must be used for sample transfer,
resulting in relatively long turnaround times.
The VTSTM design is significantly more flexible (Fig. A.4a). [5] It employs a flow-
through cryostat, in which a small volume of liquid helium is cycled through the non-vacuum
side of a Copper cold finger. The thermal mass of the cold finger is relatively small, so its
temperature can be adjusted fairly quickly by either adjusting the Helium flow rate or by
flowing current through a heating element located on the vacuum side of the cold finger.
A feedback loop is employed to control the flow of power to the heating element, allowing
us to maintain a constant temperature on the cold finger. However, the temperature of
the microscope stage increases significantly on refils of the Helium dewar, meaning that the
time available to take measurements is limited by the amount of Helium the dewar can
123
store. The microscope assembly itself consists of two sets of heat shields anchored to the
cold finger (Fig. A.4b). They shield the microscope stage, which sits inside, from radiative
heating. The cooling of the stage assembly occurs by turning a screw that pins the stage
against the heat shield. The screw will be turned in before a new sample is placed on
the stage to optimize cooling. Once the sample has thermalized, the screw is turned out,
and the microscope stage hangs from the cold finger by three springs at its perimeter. The
temperature of the microscope stage is now a balance between cooling through these springs,
and the heat leak of having electrical connections between the microscope and the outside
of the chamber (which are made with .001′′ diameter stainless steel wire). These springs
serve as the low frequency vibration isolation, with some additional magnetic damping.
This damping is provided by magnets that are attached to the microscope stage and induce
eddy currents in the Copper heat shields when the microscope stage moves. Because of the
compact construction, complete access to the entire chamber, including the mircoscope stage,
is provided by a single multimotion wobble stick. This setup allows for variable temperature
operation while maintaining good vibration isolation, and the compact construction results
in fast turnaround times.
A.2 Operation
A standard procedure was used to prepare both tips and samples in order to maximize the
probability that a good tunnel junction could be established. The goal of the tip preparation
procedure is to ensure that the STM tip is simultaneously sharp, stable, and clean before
moving on to measuring the samples of interest. This was accomplished by modifying the
tips on clean metal surfaces. In the case of the LTSTM, where the tip cannot be replaced
in situ, field emitting the Ir tip to clean Cu(111) and Ag(111) surfaces was necessary, as it
typically had significant debris at its apex from earlier experiments, or the hard oxide that
124
Optical
Table
Vacuum
Pumps
Load Lock/
Cleaving ChamberSurface Prep Tools
Flow-Through
Cryostat
Dewar
Heat Shields/Microscope
Sample
Manipulation
A B
Figure A.4: (A) This schematic diagram of the VTSTM highlights the keyparts dealing with surace preperation, vibration isolation, sample manipula-tion, and temperature regulation. (B) This is a schematic diagram of themicroscope stage, the thermal anchoring screw, and the inner heat shields,which are anchored to the cryostat. Not shown are the outer heat shields, andthe three springs used for vibration isolation(from Ref. [2]).
forms on exposure of Ir to air. A tip that has become dirty is first held close to the sample
surface with a large applied bias of +200V , which accelerates the electrons emitted by the
sample towards the apex of the tip. This causes large, sudden changes in the field emission
current (≈ 1 − 10µA), leading to a catastrophic event where the tip either crashes into the
sample (either the surface or tip melts and joins the other) or current drops to zero (the tip
melts and gets shorter, or drops off onto the surface). Afterwards, the microscope walks to
a different part of the sample, and pristine regions of the metal surface are imaged with the
new tip. Small modifications to the tip can be made by lightly poking it into the sample
surface. The procedure is repeated until a satisfactory tip results. The need for harshly
preparing tips in the LTSTM is alleviated due to its ability for insitu changing of the tip. In
order to avoid having to harshly prepare tips, commercially etched PtIr (90/10) tips were
used, as they are both sharp and do not form a hard oxide while being stored in ambient
conditions. Lightly poking the tip (by ≈ 10A) into prepared Au(111) surfaces is sufficient
to remove the water that naturally builds up on even inert metals in contact with air.
125
Each tip was then checked against a set of criteria which at first seem arbitrary, but have
led to the repeatable establishment of good tunnel junctions. First, the tunneling current is
confirmed to depend exponentially on tip-sample distance. This ensures that we really have
a vacuum tunnel junction. Second, the density of states of the metallic surface is verified to
be roughly flat, with a clearly visible surface state. This ensures that the tip itself is metallic,
and can reliably take density of states spectra. Finally, the tip is used to image step edges
running in four orthogonal directions. This ensures that the tip doesn’t have artifacts, for
example, from a double tip. The stability of the tip is tested as an accidental byproduct of
this procedure by requiring that no tip changes (and the corresponding sudden change in
current) occur while acquiring these images.
The Bi2Sr2CaCu2O8+δ samples examined in this thesis were all processed in a way to
yield reliably flat, clean cleaves. First, the samples themselves were chosen to be large
enough that the approach procedure could find the samples, but small enough that they
cleaved straight across the sample (typically 1− 2mm on a side, 0.5mm thick). Second, the
mechanical stress of cleaving a sample requires the sample to be the weakest joint in the
sample holder- sample- post sandwich. As a result, we must choose the strongest bonding
agents possible. We chose H20E from Epotek to join the sample to the sample holder, as it
is the strongest conducting epoxy commercially available. The cleaving post was attached to
the sample surface using TorrSeal, an insulating epoxy that is stronger than H20E. In extreme
cases, H74F from Epotek could be used for both purposes, with the electrical contacts to
the sample being painted on using H20E after mounting the sample. Both joints were cured
at 100C, which, although it might result in some oxygen loss, produces significantly better
cleaves than lower temperature cures. In addition, when mounting the sample, care must be
taken not to pot it with epoxy. Finally, because the best cleaves come from samples which
already have a flat surface to attach the cleaving post, samples must often be pre-cleaved a
few times on the bench. The sample is then inserted into the vacuum chamber and cleaved
126
at a base pressure of 1e−9torr or better. This ensures nothing collects on the sample surface
when a new layer of the crystal is exposed in the cleaving process. Finally, the samples are
allowed to thermalize on the STM stage for a day. This is required because the sample is
cooled only by the legs of the microscope stage, which conducts heat very poorly. Even a
10mK change in sample temperature over the period of a day shows up as significant drift
of the tip-sample junction during open-feedback loop measurements, such as spectroscopic
measurements, and makes them difficult to take accurately.
The final step in setting up an experiment is the microscope approach. This sensitively
depends the voltages supplied to the piezos, and the parameters for the feedback loop.
Experimentally, the voltages supplied to the leg piezos were the smallest ones that walked
the sample towards the tip without significantly skewing the ramp off-axis, which can cause
the tip to miss the sample and find the sample holder. Notably, the reliability of walking is
determined by the ability to unstick the balls at the end of the leg piezos from the sample
holder. The minimum vertical deflection d (Fig. A.2) is given by the force balance equation
kd > µmg, where k = Y A/h ≈ .05N/nm is the spring constant for the piezo, A = 7mm2
is the cross sectional area of the piezo, h = 12mm is the height of the piezo, m ≈ 20g is
the mass of the sample holder, µ ≈ 10 is the coefficient of sticking, and g = 9.8 ms2 is the
gravitational constant. In practice, the minimum deflection that will serve as the threshold
for reliable step motion is 400A, or about 80V for piezos with a sensitivity of 5A/V . Here
we have assumed that the slew rate of the power supply causing this motion, which will
be set by the output resistance of the supply coupled with the capacitance of the piezo, is
sufficiently fast for this force to be considered an impulse (10V/µs was seen to be adequate).
Once the vertical motion of the tube is set to be larger than the threshold, almost any
horizontal step can be taken. The angle of the ramp translates this horizontal motion into
a shortening of the tip-sample distance. Even when done carefully, this distance forms a
distribution whose width is of the order of the step size itself. Because the full extention of
127
the scan piezo must be larger than this distance, lest the tip crash into the sample, relatively
small horizontal deflections were used (500 − 1000A, which translates to a reduction of
the tip-sample separation by 25 ± 25A to 50 ± 50A per step). Feedback loop parameters
for the approach were selected after attempting a large number of approaches on a clean
metal surface. Parameters that extend the tip too fast result in ringing when the sample is
found, which leads to tip instability. Parameters which extend the tip too slowly lead to an
overextention of the tip, and introduce the possibility for the tip touching the sample. Both
these extremes result in debris from the tip being deposited on the approach area, and can
thus be tested for. Fortunately, both the feedback loop parameters and the voltages supplied
to the piezos need only be determined exhaustively once.
In practice, the dominant mode of failure for these experiments is an unstable tunnel
junction. The two primary cultprits appear to be the tip and the samples. Even after
preparing the tip, changes can occur at its apex, for example, from the thermal shock of
putting a sample on the microscope. We believe, more often, tip-related instabilities result
from preparing a dull tip. If the tip is dull, it’s possible to be tunneling out of two different
parts depending on the slope of the sample surface. Hence you could end up preparing one
part of the tip, and tunneling into the experimental sample with a dirty part. This possibility,
which was only circumstantially confirmed, led us to use electrochemically etched tips that
were gently prepared, instead of tips that had become dull through field emission.
Failures related to the quality of the sample surface currently account for the majority of
rate of failure. The surface must be flat, with no irregularities resulting from cleaving from
two different parts of the crystal. To a certain extent, this can be detected by simply looking
at the sample after cleaving and putting it on the microscope stage. The surface itself must
also have a contact resistivity that is less than the typical resistivity of the tunnel junction.
This too can be checked by measuring the bulk resistivity of a sample and hoping that there
is no surface effect. However, currently, the overwhelming majority of our approaches do
128
not work for a different reason. The failures derive from an interaction between the tip
and loose debris on the surface. The source of this debris does not appear to be related to
the background pressure of the chamber, as its concentration is insensitive to the pressure
under which the sample is cleaved. It also does not appear to come from outgassing as the
sample stage is warmed up when coming into contact with a room temperature sample- the
concentration is also insensitive to the temperature we take measurements at. We believe it
is an intrinsic property of the crystals themselves. Under the same conditions, some cleaves
produce cleaner surface. We are able to diagnose the quality of the samples to an extent using
high magnification phase cobntrast optical microscopy. Samples with any obvious surface
irregularities never work. However, of the ones which do not have any surface irregularities,
we have not yet been able to develop a procedure that determines which samples might
produce clean cleaves and which ones never will.
A.3 Limitations
Assuming we have established a stable tunnel junction, there are three limitations to our
ability to take good data. First and foremost are those that limit the signal to noise ratio
of the tunneling current, which can result in useless, noisy data. Then are those limiting
the resolution of the data, which can render a perfectly quiet data set useless. The final
limitation is time itself, which is currently the factor limiting us from taking even larger
data sets. Each limitation applies differently to topographic and spectroscopic data, and
will be treated separately.
The signal and noise in topographic measurements is the dc level of the tunneling current
and the peak-peak ac level on top of it, respectively. However, the feedback loop, whose
output we record as the height of the tip, has an extremely limited bandwidth (10Hz) and
thus the noise is unaffected by high frequency levels on the tunneling current. The relevant
129
low-frequency ac noise comes from two sources: noise originating from the electronics, which
is independent of the setpoint of the feedback loop, and noise originating from vibrations,
which scales with the tunneling current.
Electronic noise in topographic measurements has two sources. High gain transimpedence
amplifiers (typically 1 × 1010V/A) have broadband noise whose amplitude increases with
the magnitude of the shunt capacitance on their input. The relevant capacitance is the
capacitance between the current line to ground and the bias lines to ground (≈ 100pF ).
Measured on the bench, with a 120pF shunt capacitance, the total noise at a bandwidth
of 1kHz is measured to be 2.5pArms for the DL Instruments Model 1211 on the 109V/A
setting (used in Chapters 3 and 5 of this thesis), 0.9pArms for the same amplifier on the
1010V/A setting (not used), and 1.1pArms for the Femto LCA-1K-5G (Chapter 4). The
Femto amplifier has the dual advantages of shorter cable length and fewer connections,
which reduces input capacitance from ≈ 120pF to ≈ 70pF , and results in better performance
(0.6pArms) than the DL Instruments amplifier. However, this noise has most of its weight at
frequencies outside of the bandwidth of the feedback loop, and can be ignored (only 0.1pArms
or less at the frequncies of interest). The limiting electronic noise actually arises from the
triboelectric coupling of vibrational noise into the bias and tunneling lines (≈ 1.5pArms).
The result is ≈ 4.2pA of peak-peak low frequency noise that does not depend on how much
dc current passes through the amplifier.
Although vibrations result in small changes of the tip-sample separation, the tunneling
current varies exponentially with tip-sample separation, making vibrations a major source
of topographic noise. The vibration isolation scheme has three primary components: an
acoustical room which attenuates air-borne sound above ≈ 100Hz, an optical table which
attenuates ground-borne vibration above ≈ 100Hz, and the springs suspending the micro-
scope stages which attenuate vibrations above ≈ 5Hz. The relatively well-isolated STMs
used here had vibrational noise at three primary frequencies: ≈ 5Hz and ≈ 42Hz arising
130
Frequency (Hz)
Curr
ent (r
ms, pA
)
Building Vibrations
Roughing Pump
Air handling system
0.0
0.2
0.4
0.6
0.8
1.0
0 20 40 60 80 100 120 140 160
Figure A.5: This shows the power spectrum of the tunneling current acquiredin open-feedback loop conditions with a I = 1nA, V = 100mV junction onAu111. The highest three noise peaks are labeled. The pumps are normallyturned off. The total vibrational noise was ≈ 10%.
from the vibrations of the building, and ≈ 2kHz arising from the resonance frequency of
the piezoelectric tubes (Fig. A.5). These accounted for an ac vibrational noise that was
typically 10% (peak-peak) of the dc tunneling current, with most of the power being at
low frequencies. Notably, additional changes in the tip-sample separation arising from the
voltage supplied to the piezos, which is heavily filtered (to 10Hz for the legs, and 1kHz for
the scanner), is not an issue for our experiments.
To maximize the signal to noise ratio on the tunneling current itself, the setpoint is set
such that the proportional noise, from vibrations (10%), is as large as fixed noise, from
triboelectric contributions (≈ 4.2pA). For a dc tunneling current of 50pA, we should have
6.5pA =√
(0.1 ∗ 50pA)2 + (4.2pA)2 of peak-peak ac current noise, yielding a maximum
height noise of 1A ∗ log(1 + 6.5pA/50pA) = 53mA in the topographs (typical atomic corru-
gations on Bi2Sr2CaCu2O8+δ are of the size 200mA). Increasing the tunneling current can
reduce the relative contribution of the triboelectric noise, and effectively reduce the height
131
noise to 40mA. However, decreasing the tip-sample separation by increasing the tunneling
current often results in increased interaction of the tip with impurities on the sample sur-
face. These instabilities show up as spikes in the tunnel current, and, if there is a change in
the tip, a sudden change of height of the topograph. Altough these spikes are tolerable for
topographic data, they limit our ability to take spectroscopic measurements.
The role of noise in conductance measurements is much more complicated. In a typical
conductance measurement (Fig. A.6), an AC lockin provides a clean sin wave at a specific
frequency that is applied to the sample bias (EG&G models 124A and 5209 were used in
Chapters 3 and 5, and in Chapter 4, respectively). The AC response of the tunneling current
is fed to the transconductance amplifier, which goes to the lockin. The lockin measures the
amplitude of the wave at a specified phase relative to the reference wave using a demodulation
sin wave. The phase must be set to measure only the resistive response, and not the capacitive
response, of the tunnel junction. This can be accomplished by offseting the tip out of
tunneling range, such that the tip is fully extended and no tunneling current results, and
then opening the feedback loop. In this condition, the reponse is fully capacitive, and the
lockin phase can be set to measure the amplitude of the ac response of the tunnel junction out-
of-phase with the capacitance (i.e., the conductance). The approximate signal for tunneling
conductance measurements is then dIsignal(V ) = 2.8 ∗ dVrefI/V peak-peak, where dVref is
the rms amplitude of the bias modulation, I is the DC tunneling current, and V is the
bias. This signal, typically of order 2.8pA peak-peak, is more sensitive to noise than the
topograph, where the signal level was over an order of magnitude larger. On the other hand,
only the noise sources with a frequency close to that of the sin wave contribute. This noise
again has two primary sources: electronic, and vibrational.
The primary source of electronic noise can come from the process of making the lockin
measurement itself. The input to the lockin amplifier has a strong band-pass filter on it,
which filters the ac response of the tunnel junction to a 3dB bandwidth of the order of
132
dV
Sample
Tip
+
V
A/V
Amplifier
I+dI
Bandpass
φ
Phase Shifter
X
Low-pass
Lockin Amplifier
Figure A.6: This schematic diagram highlights the five main functional partsof a lockin amplifier: the ac voltage source, the phase shifter, the bandpassinput stage, the multiplier, and the low-pass output stage.
10Hz. To a close approximation, the input can thus be treated as A sin(2πft). To measure
the amplitude of this wave, it is multiplied by a demodulating sin wave with a phase shift,
giving A sin(2πft)×2 sin(2πft+φ), which, after simplifying, is A× (cos(φ)+cos(4πft+φ)).
The first of these terms is a dc level corresponding to the amplitude of the input wave at a
phase φ, selected to measure the contribution from the conductance. The second is an equally
large 2f contamination of this measurement, consisting of a component A cos(φ) cos(4πft)
whose amplitude is also proportional to the conductance, and a component A sin(φ) sin(4πft)
whose amplitude is proportional to the capacitance. This capacitance arises from capacitive
coupling between the tip and bias lines, and is typically ≈ 1pF . At a frequency of 500Hz, it
has an amplitude of f ∗ 1pF = 0.5pArms per mV of dVref , which is equal to the magnitude
of both the dc lockin signal from the conductance and the 2f noise from the conductance
for an I/V = 0.5nS. The amplitude of the 2f noise cannot be decreased by increasing
dVref , as the amplitude of the dc conductive response, the 2f conductive response, and the
capacitive response are all proportional to dVref . It also cannot be decreased by changing
the dc current or bias of the tunnel junction itself, as the 2f conductance noise is equal in
magnitude to the lockin dc conductance term. These 2f components can only be filtered
using the output stage of the lockin, a 4th order low-pass filter with time-constant τ and a
133
12dB/octave rolloff. This leads to an attenuation of the 2f contaminent by a factor of 10,
100, 300, and 5000 for time constants of 5mS, 10mS, 30mS, and 100mS, respectively. The
effectiveness of this filter cannot be increased by raising f , as this increases the capacitive
response. Although this analysis is specific to analog lockins, digital lockins suffer from
similar high-frequency contamination that depends on how they take Fourier Transforms.
Often the amplitude of the contamination is smaller, but it is distributed over a wider range
of frequencies, including low frequencies, making it harder to filter.
Assuming we have chosen a reasonable time constant, the primary source of electronic
noise in lockin measurements should come from noise on the input stage near the lockin
frequency, which will appear as low-frequency beats on its output. The frequency must thus
be chosen at a point where the background noise levels are featureless, and small. The lockin
frequency of 500Hz is far away from the charcteristic frequencies for triboelectric noise, and
thus represents a good choice. How much noise near the lockin frequency contributes to the
measurements depends sensitively on the bandwidth of the lockin. The limiting bandwidth
is determined by either the time constant of the output stage (a 4th order filter with a
bandwidth set by 1/τ), and or the band-pass filter on the input (a 2nd order filter with a
Q = 10, or a 4th order filter with a Q = 2 depending on the input mode used). The use of
filters with Q values much greater than 10 introduces phase shifts near the lockin frequency
that destabilize the lockin. Assuming noise near these frequencies is flat, these sharp filters
with long tails can be approximated by a hypothetical filter with a perfect cutoff, and an
effective bandwidth (Fig. A.7). For a lockin frequency of 500Hz, the calculated effective
bandwidth is 386Hz for the 4th order input filter, and 300Hz for the 2nd order input filter.
Both these are dwarfed by the effective bandwidths of the output stage- 300Hz for the
τ = 5mS setting, 156Hz for the τ = 10mS setting, 52Hz for the 30mS setting, and 16Hz
for the 100mS setting.
The sources of electronic noise near the lockin frequency are intrinsic to measuring tun-
134
Am
pli
tud
e
Frequency
BW
EBW
ω0
Figure A.7: The effective bandwidth of a bandpass filter tends to be quite abit larger than its 3dB bandwidth. One can be calculated from the other byintegrating the absolute value of A(iω) = ω0× iω/((iω)2 +BW ∗ (iω)+ω2
0)N
over all frequencies ω. In this expression, N refers to one half the order of thefilter, ω0 is the center frequency, and BW is the 3dB bandwidth (= ω0/Q,where Q is the quality factor of the filter).
neling currents. One comes from the noise intrinsic to tunneling discrete charges. So-called
shot noise has a contribution of 4fArms/√
Hz =√
2q ∗ 50pA for a 50pA dctunneling current.
The other cultprit is the amplifier. The measured power spectral density of noise near 500Hz
on the bench with the appropriate shunt capacitance are 3.2fArms/√
Hz for the Femto am-
plifier, 4.3fA/√
Hz for the DL Instruments on a gain of 1010V/A, and 10fA/√
Hz for the
DL Instruments on a gain of 109V/A. These two sources will add in quadrature to give about
5fArms/√
Hz of noise. For the τ = 5mS setting, this corresponds to 250fA (peak-peak) of
broadband low-frequency noise on the output of the lockin, 175fA for the τ = 10mS setting,
100fA for the τ = 30mS setting, and 56fA for the 100mS setting. This contribution is fixed
as a function of dVref and the dc tunnel junction parameters, and can thus be minimized by
changing either quantity. It should also be noted that some tunnel junctions become plagued
by blips in the tunneling current, usually coming from either changes in the tip configuration,
or interaction of the tip with impurities on the sample surface. The spectral content of a
delta function of amplitude a is distributed equally over all frequencies, including the lockin
135
frequency. This is usually significantly larger than the lockin signal, and invalidates data
acquired within a time 100mS + 2τ of the blip. In principle, if they are infrequenct enough,
and small enough in amplitude, they can be accounted for simply measuring their spectral
density at the oscillator frequency (in fa/√
Hz) and multiplying by the square root of the
effective bandwidth to estimate their contribution to noise.
Vibration of the tunnel junction occur far away from the lockin frequency frequencies,
but still contributes noise because they change the conductance of the tunnel junction itself.
Their contribution to total noise is a flat 10% (peak-peak) of narrow-band low-frequency
noise. Like the 2f noise from the lockin, this also scales with the oscillator amplitude
and the absolute conductance, and can only be filtered out. However, it occurs at low
frequencies, and filtering is problematic becuase of the extremely long time constants that
must be employed to filter low-frequency noise.
The total effect of noise on spectroscopic measurements clearly depends strongly on the
time constant of the output circuit on the lockin. Adding the cumulative effects of noise
together, we get a peak-peak noise of
dInoise = [(7.8EBW × 2qI + SD2amp) +
(7.8dVref2
A2 × ( IV
)2 + (fC)2) +
(0.1dVref I
V)2]0.5, (A.1)
where I is the dc tunnel current, V is the dv bias voltage, dVref is the rms amplitude of the
lockin modulation at frequency f , EBW is the equivalent bandwidth of the lockin output
stage (Table A.1), A is the attenuation factor for 2f noise (Table A.1), q = 1.6e−19C, C is
the capacitance between the current and bias lines, and SDamp is the spectral noise density
in pArms/√
Hz of the amplifier near f (which should be replaced with the measured spectral
density of noise near f if the tunnel junction is unstable). For the τ = 5mS setting, the
dominant noise has a 2f component from the lockin itself and gives 40% (peak-peak) noise
136
Time Constant EBW(Hz) A
5mS 300 10
10mS 150 100
30mS 50 300
100mS 16 5000
Table A.1: The time constant on the output stage of the lockin affects thetotal noise of spectroscopic measurements by limiting the effective bandwidthof the measurement (EBW), and by attenuating 2f contamination by a factorA.
on the signal out of the lockin. Because it is invariant on the other parameters of the
measurement, this setting is almost useless. For the τ = 10mS setting, 2f noise (4% of total
signal) is dwarfed by an equally stubborn narrow-band low-frequency noise from vibrations
that gives 10% (peak-peak) noise on the lockin signal. There is an additional 175fA (peak-
peak) of broadband low-frequency noise from intrinsic sources. The latter becomes a serious
issue when total signal levels are small enough that 175fA > 10% × dIsignal, but can be
overcome by increasing dVref . For larger time constants, the influence of this noise decreases
(100fA for τ = 30mS, 56fA for τ = 100mS), and the measurements are often purely limited
by the vibration levels of the tunnel junction. These values agree within a few percent of
what was actually measured from tunnel junctions on metal surfaces on both microscopes
as measured using a HP3563A spectrum analyzer and a HP34401 voltmeter.
Beyond insuring that there is a clean enough signal to take meaningful data, we must also
make sure that we have can appropriately resolve the features we are searching for. In this
thesis, we have focused on taking spectroscopic measurements correlated with topographic
measurements, both of which can be limited by spatial resolution. Ultimately, spatial reso-
lution is limited by sources of positional error. The voltage noise on the piezoelectric scanner
is around ±2mV (peak-peak) of combined broadband noise from the power supply (filtered
137
to 1kHz) and triboelectric noise on the voltage lines, and corresponds to a positional error of
about ±10mA. This is over an order of magnitude smaller than the typical distance between
atoms, > 1A, and is therefore never an issue. The voltage noise on the piezoelectric legs is
filtered to 10Hz and contributes even less to positional noise. Finally, horizontal vibrations,
which are much smaller than vertical vibrations (which themselved are only ≈ ±50mA),
also do not contribute to our spatial resolution. In fact, spatial resolution is limited purely
by user-selected pixelation (N × N pixels) and image size (xA × xA). The limitations on
resolution in real-space are then trivial- we can resolve features as small as x/N (Nyquit
limit), and as large as x (image size).
Determining the resolution limits on Fourier transforms of STM data, however, is a more
complicated affair, although it is limited by the same two principles. These limitations are
best illustrated by atttempting to establish error bounds on our estimation of the wavelength
from the location of a peak in a Fourier transform of the data (Fig. A.8). Assuming that
the peak width (δl, in pixels) is less than the peak location (l, in pixels), the wavelength
(s ± δs) is given by
s ± δs = x/(l ± δl)
= (x/l)/(1 ± δl
l)
=x
l× (1 ∓ δl
l). (A.2)
The wavelength and uncertainty are then simply
s = x/l (A.3)
δs =x
l(δl
l+ O(
δl
l)2) = s
δl
l. (A.4)
Notably, resolution can be improved by increasing the image size, which increases l, but not
by increasing N , which only determines the shortest resolvable wavelength (expands dynamic
range). Finally, it is valuable to know the error of the measured width of the peaks in FTS,
138
which are used to estimate domain size. The full-width half-max 2δl in FTS corresponds to
a domain size of x2δl
, with an uncertainty in this quantity of x2δl±2
− x2δl
= ± x2(δl)2
.
After locating a feature in FTS, we often want to choose an optimal image size x and
pixelation N to learn about its dispersion. Assuming that a periodic feature in a spatial
density of states map has a wavelength s at one energy, and s + ds at another energy, we
want to choose x and N such that the first feature appears l pixels away from the center in
FTS and the second l + dl pixels away (Fig. A.8). This requires s = x/l and s + ds = xl+dl
.
Simplifying, the optimal choice for the image size
x =dl × s2
ds. (A.5)
The choice of N is limited by the Nyquist criterion, which requires 2x/N = s + ds or,
simplifying,
N =2s × dl
ds. (A.6)
A naive choice of dl is simply 1, the minimum to resolve the change in wavelength, but this
will lead to an uncertainty in the change in wavelength that is of the order of ds. Instead,
a better choice is one that’s large enough to observe the breadth of both wavelengths δs,
resulting from either energy broadening or domain size. Assuming we devote δl pixels to
resolving the width, we should pick
dl =ds × δl
δs. (A.7)
Here, the choice of δl should be determined by how well we need to determine the actual
width δs, and 1 is an appropriate choice. The bredth of a given peak in FTS is often the
result of energy broadening, which we discuss next.
For all spectroscopic measurements, we must also ensure that we can resolve the features
we are looking for as a function of energy. For conductance spectra taken at a fixed location
as a function of energy, thermal broadening and the lockin dVref amplitude limit our energy
139
l
2δl
l+dl
Figure A.8: STM conductance maps with periodic modulations give FourierTransforms with distinct peaks. The peak location at one energy is l, with ahalf-width of δl. The peak location at a different energy is given by l + dl.
resolution. Recall from Chapter 2 that the tunneling conductance
dI(~r, V )
dV∝
∞∫
−∞
dEρS(~r, E)d
dV[f(E + V )], (A.8)
where ρS is the sample density of states at an energy E and position ~r, and f is a Fermi
function. Thermal fluctuations affect our ability to resolve the density of states through the
derrivative of the Fermi function. This can be approximated fairly well by a Gaussian of
full-width half-max 3.5kBT (1.4mV at 4.5K, 12mV at 40K, and 30mV at 100K). Added
in quadrature is the energy broadening from the amplitude of the bias oscillation that the
lockin uses to measure dIdV
. Assuming that we have added a modulation dVrms, the resultant
energy broadening is given by the peak-peak variation in the bias- 2dVrms
√2. Taking care
that it doesn’t significantly impact energy resolution, it is often desirable to increase dVrms
in order to increase the signal-noise ratio of conductance measurements limited by intrinsic
sources, such as shot noise and the noise of the amplifier. In cases where the signal-noise
ratio is limited either by 2f noise or vibrations, increasing dVrms has little impact outside of
degrading energy resolution. In sum, the total energy resolution
±δE =√
3(kBT )2 + 2(dV 2rms). (A.9)
140
The energy resolution can also impact spatial resolution of spatial conductance maps.
These data are frequently taken at two (or more) fixed energies as a function of position
on the surface, and a Fourier Transform taken to determine the periodicity of modulations
often found on these maps. For example, modulations found in density of states maps often
arise from the elastic scattering of quasiparticles in a band, and thus the modulations have a
wavelength that changes as a function of energy (s(E)). The energy resolution ±δE will then
introduce a width to the peaks in the Fourier Transforms through this dispersion relation, in
effect reducing spatial resolution. For a given energy E (relative to EF ), an energy resolution
of δE will introduce an intrinsic breadth to the measured wavelength s of
δs =ds
dE× δE. (A.10)
Expressed in terms of the more-commonly known dispersion relation E(k),
δs =2πδE
k2
1
dE/dk. (A.11)
The ultimate limitation on the STM experiments presented here is simply time. After
checking a tip, and before cleaving a sample, the lHe dewar is refilled to its maximum
capacity. This defines the maximum amount of time (about 144 hours for the 60l dewars
we used) we have to take data from one area, as refilling the dewar requires us to walk
out of tunneling range, and then re-approach. After refilling, we cleave the sample, and let
it cool on the stage for a day (24 hours) to let it thermalize. The approach is relatively
slow, as we’re trying to move about 0.25mm in 50A steps, and can take another 24 hours.
Finally, even after we’re in tunneling range, we must typically wait another 6 hours for the
dielectric relaxation of the piezoelectric tubes to die down, as they will change the tip-sample
separation in the open-feedback loop conditions needed to take spectra. We have a seemingly
long 90 hours remaining in which to take measurements. This is more than enough for taking
topographs and spectra as a function of energy at a line of points, as done in Chapters 3
and 5 of this thesis.
141
Spatial maps of the conductance taken at several energies, on the other hand, are very
time-intensive due to the sheer volume of data involved, typically 500,000 data points or
more. Assuming we devote a time T to taking one set of data, we can take data at a number
of energies
ε = (T
N2− O)/t, (A.12)
where N is the number of pixels, O is the overhead time per pixel, and t is the time it takes
to make a single lockin measurement (Fig. A.9). The overhead time per pixel O should be
determined only by the inverse of the feedback loop bandwidth. Once we move our tip to a
new point, we must wait 100mS (the characteristic settling time for the feedback loop, which
has a bandwidth of 10Hz) to record the output of the feedback loop for the topographic
measurement. Because we must take spectra in open feedback loop conditions, we must wait
another 100mS after we take a spectra for the feedback loop to settle down again. There
is also the additional overhead of moving the tip to a new pixel (typically 5mS) and the
time to set the bias from the last point taken in the spectrum back to the scan condition
(typically 100mS). While this suggests O = 300mS, our software adds an additional 700mS
per point of overhead (only for spectral maps) to record the data to hard disk, creating a
total O = 1s. The time it takes to take a single lockin reading t should be determined by
how long it takes for the lockin to settle at a new reading when we set a new bias voltage.
For our lockin, this time is 100mS + 2τ , where τ is the time constant on the output stage of
the lockin. In addition, we must be careful to change the bias slowly enough that the lockin
doesn’t detect the change in voltage per unit time as a massive capacitance (which can cause
the lockin to rail), in effect doubling the settling time. Finally, we sample the lockin output
for the time τ . In sum, this makes the time to make a single lockin reading at one voltage
t = 200mS + 5τ . Unfortunately, our software adds an additional 50% to this time, making
t = 300mS + 7τ .
In sum, these limitations make taking data sets of conductance maps at several energies,
142
Time
Lock
in
Ou
tpu
t
Move tip
to new point
Wait for
Feedback
Loop, Record
Topograph
Ramp Bias
To Target
Wait for
Lockin reading
to Settle
Return
bias to
Scan
Condition
Re-engage
Feedback
Loop
Make Lockin
Measurement
1 2 3 4
5
6 7
Figure A.9: This is a rough, idealized diagram represnting the time it takesto take a single point in space of a spectral map- spectroscopy as a functionof position at one (or more) energies. The times 1,2,6 and 7 are part of theoverhead (O in the text)- the amount of time it takes to set up a spectroscopicmeasurement. The times 3, 4, and 5 are parts of the time it takes to take aspectroscopic measurement at one energy (t in the text).
such as presented in Chapter 4, very difficult. In Chapter 4, we attempted to track a
modulation in the density of states with a wavelength of s = 4.7a0 = 18A. Say that we want
to know whether its wavelength changes by ds = 0.2a0 = 0.76A. This forces an image size
of x = 182
0.76A = 432A. We still want to be able to see the atoms on this image, meaning
that the Nyquist limit dictates 2x/N = 3.8A or N > 228. The time constant we choose
is limited by what we want our signal to noise ratio to be, which would ideally be limited
only by vibrational noise, an almost unchangeable 10% noise level. The approximate signal
depends on the dc tunneling parameters I and V , which were chosen to be 40pA and 150mV
because they gave a reasonably small amount of noise arising from tip-impurity interactions.
The remainder of the noise arises from intrinsic sources, and has a total contribution of
100fA for τ = 30mS. In order to ensure that this contribution is smaller than that from
vibrations, we chose dIsignal = 28 ∗ 100fA = 2.8pA (vibrational noise will be 10% of this,
or 280fA). This requirement and the dc tunnel junction parameters force the choice for
143
dVref = (2.8pA × 150mV )/(40pA × 2.8) ≈ 4mVrms. These choices should yield a noise level
of about 11% total, a t = 300mS + 7τ = 500mS, and an energy resolution that is almost
purely limited by kBT (±δE = ±16mV , where the thermal limit is ±15mV ). Assuming we
spend the entire T = 90 hours = 324000s to taking a single data set, this set can contain
just ε = (324000s2562 − 1s)/(0.5s) = 7.88 ≈ 7 − 8 energies. This kind of analysis was used to
determine our choice of parameters for the data set taken in the pseudogap state in Chapter
4. However, observed signal-noise ratio was worse than 11% due to interactions between
the tip and the sample, which created spikes in the tunneling current. We should have
attempted to estimate the spectral density of this noise, and set dVref to be higher, in effect
compromising a little bit of resolution for a better signal-noise ratio.
A.4 References
[1] D. Hornbaker, Ph.D. thesis, University of Illinois at Urbana-Champaign, 2003.
[2] M. Vershinin, Ph.D. thesis, University of Illinois at Urbana-Champaign, 2004.
[3] J. Frohn, J. Wolf, K. Besocke, and M. Teske, Ref. Sci. Inst. 60, 1200 (1989).
[4] D. Eigler and E. Schweizer, Nature 344, 524 (1990).
[5] B. Stipe, M. Rezaei, and W. Ho, Ref. Sci. Inst. 70, 137 (1999).
144
Appendix B
Elastic Scattering Interference
Calculations
Throughout this thesis, we have made use of the fact that the density of states in the immedi-
ate vicinity of a scattering site reveals information about the parent electronic state which is
inaccessible from examining the density of states over pristine regions. Of particular interest
is whether an or not a defect disturbs the parent electronic state in a perturbative fashion ex-
pected for elastic scattering. [1] To accomplish this, we have compared the measured density
of states to calculations based on Green functions proposed to describe the parent electronic
state of the system. In principle, such a comparison can reveal which of the proposed states
are consistent with the observed STM density of states data. They can also reveal if the
measured data is inconsistent with an elastic scattering description on general terms. In this
Appendix, we will describe the methods we used to perform these calculations.
B.1 Formulation
The elastic scattering of quasiparticles from an impurity can be treated using perturbation
theory. [2, 3, 4, 5, 6] The Green function near an impurity G(~r, ω) is then the sum of the
145
Green function over bare regions of the sample G0(~r, ω) (Fig. B.1a) plus corrections arising
from the elastic scattering. Mathematically, elastic scattering introduces a correction of the
form∫
d2~r′d2 ~r′′G0(~r − ~r′, ω)T (~r′, ~r′′, ω)G0( ~r′′ − ~r, ω) (B.1)
where the scattering matrix
T (~r′, ~r′′, ω) = V (~r′, ω) × δ(~r′ − ~r′′) +∫
d2 ~r′′′G0(~r′ − ~r′′′, ω)T ( ~r′′′, ~r′′, ω) (B.2)
and V is the bare scattering potential. In picture terms, this correction includes three kinds
of diagrams: weak scattering (Born approximation) from a single site (Fig. B.1b), strong
scattering from a single site (Fig. B.1c), and scattering from multiple sites (Fig. B.1d).
Assuming that scattering from spatially extended impurity potentials can be treated as a
spatial superposition of scattering patterns from single defects [5], the scattering matrix can
be simplified to
T (~r′, ~r′′, ω) = V (~r′, ω) × δ(~r′ − ~r′′) + G0(~r′ − ~r′′′, ω)δ(~r′ − ~r′′′)T ( ~r′′′, ~r′′, ω). (B.3)
This expression has the conventional T-matrix form, so we can rewrite the correction to the
Green function as∫
d2~r′G0(~r − ~r′, ω)T (~r′, ~r′, ω)G0(~r′ − ~r, ω) (B.4)
where
T (~r′, ~r′, ω) = V (~r′, ω) × [G0(0, ω)T (~r′, ~r′, ω)]
= (V (~r′, ω)−1 − G0(0, ω))−1. (B.5)
The second set of diagrams must be accounted for in the unitary limit (strong scattering).
However, this is important only when considering the possibility of forming bound states.
In cases where we are interested only in the spatial distribution of electronic states at a
given energy, such as in Chapters 3 and 4 of this thesis, we only need to retain the weak
146
A B C
Figure B.1: (A) This diagram depicts the weak elastic scattering of a quasi-particle by a point defect. This diagram corresponds to making a Born ap-proximation. (B) This set of diagrams depicts higher-order scattering of thequasiparticle by the same defect. Including all of these corresponds to makingthe T-approximation for strong scattering. (C) These diagrams correspond tothe scattering of the quasiparticle off multiple defects. These cannot be ap-proximated by the linear superposition of multiple single-impurity scatteringevents, although their contribution is often negligible.
scattering term T (~r′, ω) = V (~r′, ω) (the Born approximation). The total Green function
near the impurity now has the simple form
G(~r, ω) = G0(~r, ω) +∫
d2~r′G0(~r − ~r′, ω)V (~r′, ω)G0(~r′ − ~r, ω), (B.6)
where V is the scattering potential.
One effect of introducing an elastically scattering defect into a material is to produce
modulated patterns in the density of states. In the language of Green functions, the den-
sity of states is n(~r, ω) = − 1πImG(~r, ω). Substituting our Born approximation into this
expression,
n(~r, ω) = − 1
πImG0(~r, ω) +
∫
d2~r′G0(~r − ~r′, ω)V (~r′, ω)G0(~r′ − ~r, ω), (B.7)
where the first term will be spatially homogeneous because the electronic wavefunctions have
random phases, but the second will contain modulations because the defect pins the phase of
these wavefunctions. The modulated part of the density of states, which arises from elastic
147
scattering, has the form
δn(~r, ω) = − 1
πIm
∫
d2~r′G0(~r − ~r′, ω)V (~r′, ω)G0(~r′ − ~r, ω), (B.8)
in real space, and
δn(~q, ω) = − 1
πImV (~q, ω)
∫
d2~kG0(~k, ω)G0(~k + ~q, ω) (B.9)
in Fourier Transform Space (FTS). The second of these forms is particularly useful because
the density of states modulations arising from elastic scattering now contains only two terms:
one term
V (~q, ω) =∫
d2~rei~q·~rV (~r, ω) (B.10)
that is the form factor for the scattering potential, and a term
Λ(~q, ω) =∫
d2~kG0(~k, ω)G0(~k + ~q, ω) (B.11)
which contains all of the wave-interference information.
B.2 Calculation
The challenge is then to calculate the wave-interference information Λ present in the density
of states modulations arising from elastic scattering. The structure factor V acts as a filter of
the wave-interference information, and can be ignored for the purposes of the calculation, as
it only serves to mask the set of information which could be present. The wave-interference
term is complicated to calculate in FTS, as it involves a convolution integral, but is simple
to calculate in real-space, where it reduces to a product Λ(~r, ω) = G0(~r, ω)2. Because most
Green functions are specified in FTS, the strategy will then be to calculate
δn(~q, ω) = − 1
πImFFT [IFFT [G0(~k, ω)]2], (B.12)
148
where FFT denotes a forward fast Fourier transform, IFFT denotes an inverse fast Fourier
transform, G0 is the unperturbed Green function, and V has been taken to be an all-pass
filter.
The calculation of the density of states modulations arising from elastic scattering thus
boils down to a precise calculation of the Green function on a lattice and a few fast Fourier
transforms. However, care must be taken to perform the calculation in Equation B.12 on
a large lattice. Artifacts from the discrete sampling of the Green function, which contains
singularities in FTS, can contaminate the result (for example, Ref. [7]). The best approach
is to make the matrices in the calculation as large as possible. The calculation will use 3
equally sized matrices which will each occupy 1/4 of the RAM on your system, with the last
1/4 being devoted to the operating system. To determine the size of each of the matrices,
recall that double-precision complex numbers take 16 bytes of memory to store (usually
double in interpreted languages, such as MATLAB). The size of each dimension (n × n) of
each of the matrices should thus be n <√
RAM(bytes)/(16 × 4). On a computer with 1GB
of RAM, 2048×2048 matrices of double precision complex numbers should be used. We will
illustrate the procedure using MATLAB code, as it is fairly easy to understand and translate
into other languages.
The most time-intensive part of the calculation is simply to calculate the Green function
on the n × n lattice. Because we will want to repeat this calculation at several energies, it
pays to calculate quantities which will not change as a function of energy and keep them
stored in memory. This is the reason why we divide the available RAM into four equally
sized matrices. To determine which quantities should only be calculated once, consider the
fairly general form for the Green function G(~k, ω) = (ω − ε(~k) − Σ(~k, ω))−1, where ω is the
energy, ε(~k) is the band structure determined from ARPES, and Σ(~k, ω) is a self-energy that
encapsulates correlations in the system. Both ~k (of size 1 × n) and ε(~k) (size n × n) are
always energy-independent, and some parts of Σ(~k, ω) (size n × n) often are. Because ε(~k)
149
often involves trigonometric functions of ~k, we start by storing the cosines of ~k instead of ~k
itself:
size=2048;
for n=1:size,
k=4*pi*(n-1)/(size)-2*pi;
cosk(n)=cos(k);
end;
Here, we have taken ~k values between (±2π,±2π) in order to make sure our calculation
spans enough of ~k-space to not ignore Umklapp processes. The two-dimensional matrix of
ε(~k) can now be calculated:
for n=1:size,
for m=1:size,
cos2kx=2*cosk(n)ˆ2-1;
cos2ky=2*cosk(m)ˆ2-1;
ek(n,m)=120.5-595.1/2*(cosk(n)+cosk(m))+ 163.6*(cosk(n)*cosk(m))
-51.9/2*(cos2kx+cos2ky) + 51*(cos2kx*cos2ky)
-111.7/2*(cos2kx*cosk(m)+cosk(n)*cos2ky);
...
Here, we have taken ε(~k) to have the Bi2Sr2CaCu2O8+δ band structure. [8] Finally, we
should also store whatever parts of Σ(~k, ω) are energy-independent. For a superconductor,
Norman et al. [9] have proposed a phenomenological Σ(~k, ω) = −iδ+∆(~k)2/(ω+ε(~k)+ i0+).
In this expression, ∆(~k) = ∆0/2(cos kx − cos ky) is energy independent, and should also be
calculated once and stored for future use:
...
150
Dk(n,m)=45/2*(cosk(n)-cosk(m));
end;
end;
We are now ready to calculate the density of states for a range of energies (Eq. B.12).
Begin by calculating Σ(~k, ω) (as a scalar), and from it, G0(~k, ω) (two-dimensional matrix):
delta=2.0
for en=-60:60,
for n=1:size,
for m=1:size,
sigma=-j*delta+Dk(n,m)ˆ2/(en+j*delta/10+ek(n,m));
G0(n,m)=1/(en-ek(n,m)-sigma);
end;
end;
...
Next, calculate the inverse Fourier transform of G0, and square the result for all points in
real-space before transforming back into FTS:
...
G0=ifft2(G0);
for n=1:size,
for m=1:size,
G0(n,m)=G0(n,m)ˆ2;
end;
end;
G0=fft2(G0);
...
151
Finally, we can calulcate the power spectrum of δn(~q, ω), which we will do in the variable
’G0’ to save RAM:
...
for n=1:size,
for m=1:size,
G0(n,m)=-imag(G0(n,m))ˆ2;
end;
end;
We have, again, assumed the structure factor V to be an all-pass filter, although if we had
not, it should be applied inside the nested loops before calculating ’G0’ in the above code.
Finally, write the resulting elastic scattering interference image δn(~q, ω) to disk and move
on to the next energy:
...
mi=min(G0(:));
ma=max(G0(:));
G0=(G0-mi)/(ma-mi);
imwrite(G0,strcat(’c:\calc\’,int2str(en),’.tif’),’tif’);
end;
The procedure outlined here calculates the scattering interference contribution to the
density of states efficiently enough to be useful in most practical situations. With an Athlon
XP 2000MHz CPU and 1GB of RAM, and using MATLAB 6.1 in Linux, the calculation of
the power spectrum of δn(~q, ω) on a 2048×2048 grid takes about 2 minutes per energy, with
a total overhead of 2 minutes. Speed can be improved by a factor of about 4 by vectorizing
the calculations. The speed can be further improved by about a factor of 2 using a compiled
152
(0,0) (0,2π)
(2π,2π)
Figure B.2: This is the imaginary part of the Green function described in thetext (ω = −20mV ). It was calculated between kx = −2π to 2π and ky = −2πto 2π. However, we need only calculate the part in the box, and then usesymmetry transformations to copy (instead of calculate) the rest of the Greenfunction.
language such as Fortran. Another small increase in speed, and a hefty decrease in memory
footprint, can be attained by taking advantage of the fact that Green function obeys various
symmetries (e.g. four-fold symmetry) in most cases (Fig. B.2). This important improvement
was not included in the above code because it makes the code significantly harder to read.
In sum, the calculations presented in this thesis were all performed on 4096× 4096 grids for
81 energies (−80mV to 80mV ) in a total time of less than 90 minutes for a single Green
function. The complete Fortran code used to generate the plots in Chapter 3 is included
here (compiled using the Intel Fortran Compiler and the Intel Math Kernel Library with
ifl -G7 -O3 -w95 gf.f mkl c.lib). In particular, notice that we have made use of symmetry
to decrease the memory footprint, and that we have vectorized the calculation to improve
speed.
Contents of File: gf.f
153
subroutine csymm(orig,mat,n)
c This routine duplicates -2pi to -pi part of k-space, stored in
c orig, to occupy the space -2pi to 2pi in mat.
complex*16 orig(n/4,n/4)
complex*16 mat(n,n)
integer i,j
do i=1, n/4
do j=1,n/4
mat(i,j)=orig(i,j)
enddo
enddo
do i=n/4+1, n/2
do j=1, n/4
mat(i,j)=mat(n/2+1-i,j)
enddo
enddo
do i=1, n/2
do j=n/4+1, n/2
mat(i,j)=mat(i,n/2+1-j)
enddo
enddo
do i=1, n/2
do j=n/2+1,n
mat(i,j)=mat(i,j-n/2)
154
enddo
enddo
do i=n/2+1, n
do j=1,n
mat(i,j)=mat(i-n/2,j)
enddo
enddo
return
end
subroutine matrix2pgm(mat, x, n)
integer x
double precision mat(x,x)
integer n
integer i,j,k
character(len=20) name
double precision max, min
integer val
write (name, 20) n
20 format(’test’,i3.3,’.pgm’)
open (UNIT=1, file=name)
write (1,’(A2) ’) ”P2”
write (1,’(i6,i6) ’) x, x
155
write (1,’(i6) ’) 255
max=mat(1,1)
min=mat(1,1)
do i=1,x
do j=1,x
if (max .lT. mat(i,j)) then
max=mat(i,j)
endif
if (min .gT. mat(i,j)) then
min=mat(i,j)
endif
enddo
enddo
do i=1,x
do j=1,x
val=int((mat(i,j)-min)/(max-min)*255.0+0.5)
write (1,*) val
enddo
enddo
close(1)
return
end
program dN
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c Constants
integer energy
parameter (energy=121)
integer k pts
parameter (k pts=4096)
double precision mw,pw
parameter(mw = -60.0d0,pw=60.0d0)
double precision PI
c Band Structure
double precision t 0, t 1, t 2, t 3, t 4, t 5
parameter(t 0=120.5d0, t 1=-595.1d0, t 2=163.6d0)
parameter(t 3=-51.9d0, t 4=-111.7d0, t 5=51.0d0)
double precision D 0, delta
parameter(D 0=45.0d0, delta=.6d0)
c variables
double precision outp(k pts/2,k pts/2)
integer n,m,e, nn, mm, status, nn2
double precision ek(k pts/4, k pts/4), Dk(k pts/4,k pts/4)
double precision cosk(k pts)
double complex sigma1, sigma2
complex*16 G0(k pts/4,k pts/4), F0(k pts/4,k pts/4)
complex*16 temmy(k pts,k pts)
double precision cos2kx, cos2ky, w
157
PI=4.0d0*datan(1.0d0)
nn=k pts
mm=k pts
nn2=k pts/2
c Set up cosines (calculationally expensive, so do once)
do n=1, k pts
cosk(n)=dcos(4.0d0*PI*dble(n-1)/dble(k pts)+2.0d0*PI)
enddo
c Set up stuff that’s energy independent, like band structure and gap
c Note in particular that we only need to calculate this between
c -2pi and -pi, not for the whole region
do n=1,k pts/4
do m=1, k pts/4
cos2kx=2.0d0*cosk(n)*cosk(n)-1.0d0;
cos2ky=2.0d0*cosk(m)*cosk(m)-1.0d0;
ek(n,m)=t 0+t 1/2.0d0*(cosk(n)+cosk(m))
c + t 2*(cosk(n)*cosk(m))
c + t 3/2.0d0*(cos2kx+cos2ky) + t 5*(cos2kx*cos2ky)
c + t 4/2.0d0*(cos2kx*cosk(m)+cosk(n)*cos2ky)
Dk(n,m)=D 0/2.0d0*(cosk(n)-cosk(m))
enddo
enddo
c Main Loop
do e=1, energy
w=(pw-mw)*dble(e-1)/dble(energy-1)+mw
158
print *,e,w
c calculate G0 on a 1/4 of the BZ
G0=(dcmplx(w-ek,delta)*dcmplx(w+ek,delta/10))-Dk*Dk
F0=dcmplx(Dk,0.0d0)/G0
G0=dcmplx(w+ek,delta/10)/G0
c Now use symmetry transforms to get G0 for the whole BZ
c And then calculate dN
call csymm(G0,temmy,nn)
status=1
call zfft2d(temmy,nn,mm,status)
temmy=temmy*temmy
status=-1
call zfft2d(temmy,nn,mm,status)
do n=1,k pts/2
do m=1,k pts/2
outp(n,m)=dimag(temmy(n+k pts/4,m+k pts/4))
enddo
enddo
c Repeat for F0
call csymm(F0,temmy,nn)
status=1
call zfft2d(temmy,nn,mm,status)
temmy=temmy*temmy
status=-1
call zfft2d(temmy,nn,mm,status)
do n=1,k pts/2
159
do m=1,k pts/2
outp(n,m)=outp(n,m)+ dimag(temmy(n+k pts/4,m+k pts/4))
enddo
enddo
c write dN to file
outp=outp*outp
call matrix2pgm(outp,nn2,100+e)
enddo
stop
end
B.3 References
[1] S. Misra, M. Vershinin, P. Phillips, and A. Yazdani, Phys. Rev. B in press (2004).
[2] Q.-H. Wang and D.-H. Lee, Phys. Rev. B 67, 20511 (2003).
[3] L. Capriotti, D. Scalapino, and R. Sedgewick, Phys. Rev. B 68, 14508 (2003).
[4] M. Salkola, A. Balatsky, and D. Scalapino, Phys. Rev. Lett. 77, 1841 (1996).
[5] L. Zhu, W. Atkinson, and P. Hirschfeld, Phys. Rev. B 69, 60503 (2004).
[6] S. Misra et al., Phys. Rev. B 66, 100510 (2002).
[7] C. Bena, S. Chakravarty, J. Hu, and C. Nayak, cond-mat/0405468 (unpublished).
[8] M. Norman, M. Randeria, H. Ding, and J. Campuzano, Phys. Rev. B 52, 615 (1995).
160
[9] M. Norman, M. Randeria, H. Ding, and J. Campuzano, Phys. Rev. B 57, 11093 (1998).
161
Curriculum Vitae
Degrees
1998 B.S. AMEP & Mathematics, University of Wisconsin- Madison
2004 Ph.D. Physics, University of Illinois at Urbana- Champaign
Honors
Radke Award Winner (Oustanding Physics Undergraduate at Madison) 1998
Hilldale Award Winner (Undergraduate Thesis at Madison) 1998
UIUC University Fellowship (Fellowship for 1st year of Graduate School) 1998
GAANN Fellowship (Fellowship for 2nd year of Graduate School) 1999
Bardeen Award Winner (departmental award for outstanding research by a graduate student
in condensed matter physics)- 2003
PCCM Fellow (Postdoctoral appointment at Princeton University)- 2005
Papers and Invited Talks
1. S. Misra, et al., PRB 58, 8905 (1998).
2. D.J. Hornbaker, et al., Science 295, 828 (2002).
3. S. Misra, et al., PRL 89, 87002 (2002).
4. S. Misra, et al., PRB 66, 100510 (2002).
5. S. Misra, March Meeting of the APS (2003).
162
6. M. Vershinin, S. Misra, et al., Science 303, 1995 (2004).
7. S. Misra, et al., Phys. Rev. B, in press, cond-mat/0405204 (2004).
8. S. Misra, March Meeting of the APS (2005).
163
