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arXiv:cond-mat/0609250v1 [cond-mat.str-el] 11 Sep 2006

Tl2Ba2CuO6+δ Brings Spectroscopic Probes Deep

Into the Overdoped Regime of the High-Tc Cuprates

DC Peets, JDF Mottershead, B Wu, IS Elfimov, R Liang, WN

Hardy, DA Bonn, M Raudsepp†, NJC Ingle and A Damascelli

Department of Physics and Astronomy, University of British Columbia,

Vancouver, BC V6T 1Z1 Canada

† Department of Earth and Ocean Sciences, University of British Columbia,

Vancouver, BC V6T 1Z4 Canada

E-mail: bonn@physics.ubc.ca and damascelli@physics.ubc.ca

Abstract. Single-particle spectroscopic probes, such as scanning tunneling and

angle-resolved photoemission spectroscopy (ARPES), have provided us with crucial

insights into the complex electronic structure of the high-Tc cuprates, in particular

for the under and optimally doped regimes where high-quality crystals suitable for

surface-sensitive experiments are available. Conversely, the elementary excitations on

the heavily overdoped side of the phase diagram remain largely unexplored. Important

breakthroughs could come from the study of Tl2Ba2CuO6+δ (Tl2201), a structurally

simple system whose doping level can be tuned from optimal to extreme overdoping

by varying the oxygen content. Using a self-flux method and encapsulation, we have

grown single crystals of Tl2201, which were then carefully annealed under controlled

oxygen partial pressures. Their high quality and homogeneity are demonstrated by

narrow rocking curves and superconducting transition widths. For higher dopings, the

crystals are orthorhombic, a lattice distortion stabilized by O interstitials in the TlO

layer. These crystals have enabled the first successful ARPES study of both normal

and superconducting-state electronic structure in Tl2201, allowing a direct comparison

with the Fermi surface from magnetoresistance and the gap from thermal conductivity

experiments. This establishes Tl2201 as the first high-Tc cuprate for which a surface-

sensitive single-particle spectroscopy and a comparable bulk transport technique have

arrived at quantitative agreement on a major feature such as the normal state Fermi

surface. The momentum dependence of the ARPES lineshape reveals, however, an

unexpected phenomenology: in contrast to the case of under- and optimally-doped

cuprates, quasiparticles are sharp near (π, 0), the antinodal region where the gap is

maximum, and broad at (π/2, π/2), the nodal region where the gap vanishes. This

reversed quasiparticle anisotropy past optimal doping, and its relevance to scattering,

many-body, and quantum-critical phenomena in the high-Tc cuprates, are discussed.

Submitted to New Journal of Physics (September 4, 2006).

Tl2201 Brings Spectroscopic Probes Deep into the Overdoped Regime

2

1. Introduction

The cuprate superconductors can be tuned through a remarkable progression of states

of matter by doping charge carriers into the CuO2 planes [1]. The most generic feature

of this tuning is a sequence from an antiferromagnetic insulator, through a doping range

where the superconducting critical temperature builds to a maximum (optimal doping)

and then dies away again at higher doping, in the so-called overdoped regime (Fig.1).

This sequence contains three separate touchstones for an understanding of the cuprates:

the Mott insulator, which is known to result from very strong electron-electron repulsion

and is antiferromagnetically ordered, the d-wave superconductor, and the overdoped

metal. The latter is widely believed to exhibit less exotic normal-state properties

and might be understood through Fermi liquid theory. Over the last two decades, a

great deal of experimental work has been done on undoped, underdoped, and optimally

doped cuprates, aiming at elucidating the connection between the antiferromagnetic

insulator and high-Tc superconductor (HTSC), and the role of electronic correlations in

the emergence of superconductivity. However, the testing of Fermi liquid theory in the

overdoped regime has been severely hampered by a lack of compounds that can actually

be doped to this high level and are suitable for a wide range of experimental techniques.

In the struggle to understand high-Tc superconductivity, the quest for better

materials to study remains key. Researchers want samples that can be grown

very cleanly, can be doped over a very wide range of the phase diagram, and are

suitable for a variety of bulk and surface measurements. Surface-sensitive techniques

in particular, such as angle-resolved photoemission spectroscopy (ARPES) [2, 3]

and scanning tunneling microscopy (STM) and spectroscopy (STS) [4], introduce

additional constraints with regard to sample requirements. These highly sophisticated

spectroscopic probes can be used to obtain information on the single-particle excitation

spectrum of a solid — i.e. the spectral function A(k,ω) that can be calculated in terms

of the electron Green’s function starting from the microscopic many-body Hamiltonian

of the system [5] — but only if the material can be cleaved to expose a well-defined and

stable surface (i.e. which does not reconstruct, obscuring the bulk electronic structure).

A summary of the approximate doping range explored by ARPES on different HTSC

families is given in Fig.1. On the undoped side of the phase diagram (p=0), the parent

compounds Ca2CuO2Cl2 and Sr2CuO2Cl2 yield excellent cleaved [001] faces that have

allowed ARPES studies of the electronic structure of the correlated antiferromagnetic

insulator and of the magneto-elastic interactions associated with the propagation of a

single hole in the two-dimensional spin background [6, 7, 8]. Via hole doping, one can

drive these materials through a metal-insulator transition and enter the superconducting

dome (in this paper we will focus on the hole-doped side of the phase diagram).

For instance, recent STM, STS [9], and ARPES [10] experiments on the underdoped

superconductor Ca2−xNaxCuO2Cl2 have suggested the existence of new electronic

ordering phenomena in the doping region between the insulator and the superconductor,

possibly in the form of glassy electronic behavior or a surface-nucleated phase transition

Tl2201 Brings Spectroscopic Probes Deep into the Overdoped Regime

3

Figure 1. Generic temperature-doping phase diagram for both electron (p < 0) and

hole-doped (p>0) cuprates. The doping range actually explored by ARPES for those

material families more extensively studied is indicated by the corresponding arrows.

as more conventional long-range charge order was not observed in scattering studies [11].

Near optimal doping, where the cuprates exhibit their highest superconducting critical

temperatures, YBa2Cu3O7−δ stands out as the cleanest material and has been used

extensively in bulk-sensitive studies of the normal and superconducting properties, such

as the symmetry of the order parameter [12, 13, 14], superfluid density [15], charge and

thermal transport [16], low-energy electrodynamics [17, 18], vibrational and magnetic

excitation spectra [19, 20]. Unfortunately, this material is complicated by the presence

of CuO-chain layers and does not have a neutral [001] cleavage plane for surface-sensitive

techniques. In particular, YBa2Cu3O7−δ cleaves between the chain layer and the BaO

layer. BaO surfaces have had the nearest source of dopants (the CuO chains) removed,

so that their hole-doping is uncertain. STS shows that the CuO chain surfaces are

characterized by prominent surface density waves [21] and differ substantially from the

bulk. They exhibit surface states, as seen in ARPES [22], and possess unavoidable

doping disorder. These problems have severely limited the availability of single-particle

spectroscopy data from ARPES [23, 24], STM, and STS [21, 25].

Extensive ARPES investigations have been performed on La2−xSrxCuO4 [26], over

a broad doping range extending from the undoped insulator (x = 0) to the overdoped

superconductor (x=0.22; note that x=p for this compound). This system, however, is

affected by severe cation disorder associated with La-Sr substitution right next to the

CuO2 plane, as well as lattice, charge, and magnetic instabilities, which might provide

an explanation for why the maximum Tc in this compound is suppressed with respect to

other single-layer materials [27]. In turn, many of the key bulk-sensitive measurements

requiring perfect crystals and long mean free paths could not be performed for this

Tl2201 Brings Spectroscopic Probes Deep into the Overdoped Regime

4

material family. In addition, neither STM nor STS experiments have been successful so

far on cleaved La2−xSrxCuO4, probably because both the CuO2 and La1− x

2 Srx

2 O surfaces

are polar, with a charge zCuO2 = −(2−x) and z(La,Sr)O = (1−x/2), and thus critically

unstable. To date, the most rich and comprehensive spectroscopic information on high-

Tc cuprates from both STS and ARPES have been obtained on the Bi compounds,

and in particular Bi2Sr2CuO6+δ and Bi2Sr2CaCu2O8+δ [2, 3, 4]. However, similar to

La2−xSrxCuO4, this family has greater problems with disorder than YBa2Cu3O7−δ [27],

leaving a disconnection between materials for which we have the most information. This

dilemma interferes with strict tests of the theory of high-Tc superconductivity, in which

one must connect the single particle excitations to the other physical properties and

dynamic susceptibilities of the cuprates [28].

A further problem in the study of HTSCs is that, due to the high Tc values, the

physical properties of the underlying normal state can be probed only at relatively high

temperatures. This often makes the detailed interpretation of the experimental results

more complex and less informative (e.g., charge and thermal conductivity, Hall number,

magnetoresistance, etc.), or can even make some experiments completely unfeasible

(as in the case of the de Haas-van Alphen effect). Alternatively, at least for certain

techniques, one can access the low-temperature normal-state properties by suppressing

superconductivity via the application of an external magnetic field. Once again however,

due to the very high Tc and in turn the extraordinarily large value of the second critical

field Hc2, this approach is precluded near optimal doping on the cleanest cuprates.

Extreme overdoping is a means of reducing both Tc and Hc2, so that one can study both

the superconducting and underlying normal states, but here most samples tend to be less

clean because of the need to dope with large concentrations of cation impurities, such

as Sr in La2CuO4 and Pb in Bi2Sr2CuO6+δ [27]. In this case it can be unclear whether

changes observed in the material’s properties are attributable to the increased doping,

or the increased scattering and/or other extrinsic effects. Due to these material and

experimental limitations, the underlying normal state of HTSCs, the heavily overdoped

side of the HTSC phase diagram, and the metallic state that emerges beyond the end

of the superconducting dome have remained so far largely unexplored (Fig.1).

The most promising compound for investigating the deeply overdoped regime is

the single-layer system Tl2Ba2CuO6+δ (Tl2201, see Fig. 2), whose natural doping range

varies from optimal to extreme overdoping as one increases the oxygen content δ.

Tl2201 has two extremely flat CuO2 planes located far apart from each other in the

body-centered unit cell; there are no complicating chain layers (as opposed to the

Y-based cuprates), or bilayer band-splitting effects (as opposed to Bi2Sr2CaCu2O8+δ

and YBa2Cu3O7−δ). Like La and Bi-based systems, Tl2201 can exhibit a cation non-

stoichiometry but this consists in excess Cu substituting Tl in the Tl2O2 double layer,

farther removed from the CuO2 planes than the Bi-Sr or La-Sr substitutions that have

been shown to have a large impact on the superconductivity of the Bi and La cuprates

[27]. Tl2201 may be reversibly doped through interstitial oxygen and, near optimal

doping, vacancies in the Tl2O2 bilayer [29]. The ability to overdope a crystal reversibly

Tl2201 Brings Spectroscopic Probes Deep into the Overdoped Regime

5

Figure 2. (a) Unit cell of Tl2201. (b,c) Electrical resistivity and thermal conductivity

of overdoped Tl2201 (Tc ≃ 15 K), plotted vs. temperature for different values of an

external magnetic field applied normal to the CuO2 planes (after Ref. [34]). From the

T =0 K intercepts of ρ(T) and κ(T)/T, the ratio ρ0κ0/T =0.99±0.01L0 was calculated

(L0 =2.44 × 10−8 WΩK2 is Sommerfeld’s value for the Lorenz ratio L≡κ/σT), which

indicates that the Wiedemann-Franz law is obeyed in overdoped cuprates although the

normal-state ρ(T) is dominated by a non Fermi-liquid like T-linear term (see inset).

from optimal doping to the full suppression of superconductivity, by changing its oxygen

content alone, is a fortuitous property of Tl2201, although this has been successfully

demonstrated over the whole doping range only on ceramic samples.

A variety of bulk properties have already been studied in Tl2201. At optimal

doping, the pure dx2−y2 symmetry of the superconducting gap was confirmed by phase-

sensitive measurements [30], and the so-called magnetic resonant mode at 47.5 meV was

detected below Tc in neutron scattering experiments [31], the first time for a single-layer

cuprate. The high sample quality has been recently demonstrated by measurements

of angular magnetoresistance oscillations (AMRO), a bulk transport technique that

requires the long mean free paths afforded by high purity and crystallinity [32]. The

AMRO study of Tl2201 resulted in the first estimate of the normal state Fermi surface

of a cuprate superconductor from a bulk sensitive probe. The combination of charge

[33, 34] and heat [34] transport measured at low temperature on very overdoped Tl2201

in the normal state, by applying up to 13T magnetic fields to suppress superconductivity

(see Fig.2), allowed the Lorenz ratio L ≡ κ/σT to be extracted. The value obtained,

0.99 ± 0.01 L0, allowed one to conclude that the Wiedemann-Franz law is obeyed in

overdoped cuprates: the electronic carriers of heat are fermionic excitations of charge

−e, suggesting that the low-energy excitations in overdoped cuprates might indeed be

described in terms of Fermi-liquid quasiparticles [34]. At variance with the hints of

Fermi-liquid-like behavior, however, the ab-plane resistivity is not purely described by

a T2 power law but is still dominated by a T-linear term (see inset of Fig 2). In

this context, the detailed study of Tl2201 with modern single particle spectroscopies

would be extremely desirable, as it might provide direct insights into the nature of the

Tl2201 Brings Spectroscopic Probes Deep into the Overdoped Regime

6

quasiparticles in both the normal and superconducting states [2, 3, 5].

Unfortunately, despite its great potential, progress on the Tl2201 system has been

slow because of difficulties in the growth of single crystals. The thallium oxide is volatile,

reactive, and highly toxic, a combination which places constraints on the crystal growth

technique. Standard growth procedures such as flux-growth in open crucibles or the use

of a floating zone image furnace would allow toxic vapors to escape into the lab and

deplete the thallium in the reaction vessel by allowing it to sublime out in a colder part of

the furnace. It is thus essential to contain the growth components within a non-reactive

enclosure so that thallium is not lost through either reaction or evaporation.

In Section 2, we report on our effort in the growth of high-purity single crystals of

Tl2201, by a copper-rich self-flux method, and in their careful annealing in a controlled

oxygen atmosphere. We will show that this material can be cleaved to expose a

clean and stable [001] surface suitable for single particle spectroscopy experiments, as

was demonstrated by the first successful study of the Fermi surface and quasiparticle

excitations of Tl2201 by ARPES [35]. The detailed results of our ARPES study of several

doping levels will be presented in Section 3, and a discussion linked to other theoretical

and experimental results will be given in Section 4. The quantitative agreement between

the ARPES and AMRO [32] determined Fermi surfaces indicates that Tl2201 may be the

ideal testing ground for finally joining together the modern single particle spectroscopies

and a host of well-established bulk probes of metals and superconductors. In particular,

Tl2201 might be the ideal cuprate for a broad-based study of the normal metal that

may underlie the superconductor in the overdoped regime, providing an understandable

anchor point in the phase diagram of the high-temperature superconductors.

2. Tl2201 Single-Crystal Preparation

Difficult to prepare even as a ceramic, Tl2201 is even more challenging to grow as a

crystal. Tl2201 melts incongruently and its crystals are grown from the high temperature

solution (melt) made of its components. Ideally, this melt would be decanted to reveal

freestanding flux-free crystals. While this technique is commonly used for growth of

complex oxide crystals, there are a few challenges particular to Tl2201 system. The

difficulties are due to the thallium oxide precursor. Around 800◦C, trivalent thallium

oxide decomposes into a monovalent oxide and oxygen gas [36]:

Tl2O3 ⇋ Tl2O+O2 ↑ .

While the oxygen gas evolved must be allowed to escape, this decomposition leads

to a more serious problem: at growth temperatures, the monovalent oxide’s vapor

pressure is of the order of 0.1atm, high enough to allow it to escape rapidly. If the

Tl2O gas leaves the crucible, however slowly, the composition of the melt will be at

best time-varying and at worst quickly devoid of thallium. The volatility of Tl2O can

also make the subsequent annealing of the crystals difficult, as the crystal surface may

slowly decompose at temperatures at which the oxygen atoms are mobile. There are

Tl2201 Brings Spectroscopic Probes Deep into the Overdoped Regime

7

several procedures that may be employed to mitigate these problems, including using

pre-reduced thallium precursors such as Tl2Ba2O5 [37], encapsulation [38], or very high

oxygen partial pressures. A further issue is that Tl2O is quite reactive, most notably

reacting with the quartz often used for encapsulation and furnace tubes.

To date, only a handful of groups have grown crystals of Tl2201. Mackenzie’s

group, using a gold lid on their crucible as a barrier to diffusion, found they could

only successfully obtain crystals within a 2◦C by 2 minute window [39, 40]. Hasegawa

et al. used reduced thallium precursors and an alumina high-pressure bomb to control

thallium loss [41, 38]. Kolesnikov et al. [42, 43, 44] do not report any special precautions

for controlling the loss of thallium. In our growth effort, as we will discuss in detail in

Section 2.1, we developed a flexible sealing scheme that permits the oxygen to escape

as it is evolved, while at the same time containing the Tl2O.

A last crucial issue with the preparation of Tl2201 single crystals, alluded to above,

is that the crystal surfaces can degrade during the oxygen post-annealing required to set

the charge carrier content, and in turn the temperature of the superconducting phase

transition. So far most work on annealing of Tl2201 has been carried out on ceramic

samples [45], but annealing of single crystals, where surface damage is more obvious,

has been less successful — only Mackenzie’s group reported success [39]. As a result,

many crystals studied to date have been unannealed, resulting in inhomogeneous oxygen

content and broad transitions (∆Tc =10 ∼20K), which can obscure many features and

add unnecessary disorder. As part of our growth effort, we have also carefully explored

the annealing of Tl2201 single crystals, succeeding in the preparation of samples ranging

from near optimal doping to very overdoped, while avoiding surface damage.

2.1. Single-Crystal Growth

For the crystal growth we employ a copper-rich self-flux technique using a barium and

copper precursor, BaCuO2 powder that was prepared from 99.999% pure BaCO3 and

99.995% pure CuO by repeated calcining under flowing oxygen. The use of carbonate-

free barium cuprate avoids both carbon poisoning from BaCO3 and the cation impurities

that may be encountered if one employs BaO2, which is only available at lower cation

purity. Carbon contamination is of particular concern in this system because there are

several known superconducting thallium-based oxycarbonates [46].

The barium cuprate powder was mixed with 99.99% pure Tl2O3 to a final cation

ratio of Tl:Ba:Cu = 1.05:1:1 [47], then loaded into a 10mL alumina crucible. The

crucible was sealed using a gold lid fixed in place with a 1kg weight. Employing this

sealing scheme, O2 gas may still escape as Tl2O3 decomposes, but the seal becomes

very effective once the contents of the crucible equilibrate at the growth temperature,

thus preventing significant loss of Tl2O (as monitored by mass-loss measurements). The

charge was heated rapidly (300◦C/h) to 935◦C, while flowing oxygen to flush out any

escaping thallium oxide, and held for 12h to equilibrate. It was cooled at −0.5◦C/h

to 890◦C, then allowed to cool freely to room temperature. The oxygen gas flowing

Tl2201 Brings Spectroscopic Probes Deep into the Overdoped Regime

8

Figure 3. (a) Photo of several resulting Tl2201 crystals on millimeter graph paper; the

orthorhombic a-axis of one crystal is marked. (b) Field-cooled magnetization curves for

representative Tl2201 crystals annealed to different hole-doping levels; the sharpness

of the superconducting transitions is an indication of the crystals’ homogeneity.

through the furnace was bubbled through sulphuric acid and water to remove any

thallium released during growth that did not condense in colder areas of the furnace.

Most resulting crystals were embedded in flux, but free-standing platelet single crystals

of 1∼2 mm2 were harvested from voids in the flux ingot. As shown in Fig.3(a), these

crystals exhibited mirror surfaces with fine curved growth steps. All characterization

and spectroscopy reported here was performed on flux-free, void-grown single crystals.

The as-grown samples typically have superconducting transitions of 5∼10 K (onset)

and several Kelvins wide, a width comparable to other groups’ as-grown samples. As-

grown Tcs are determined by the cation substitution level, crystal dimensions, cooling

rate, and the atmosphere around the crystals, the last two parameters being coupled

through the equation of state for oxygen gas. Since as-grown superconducting transitions

are typically quite broad and not homogeneous throughout the crystal, it is crucial that

the crystals be annealed before they are studied. The crystals were annealed under

controlled oxygen partial pressures at temperatures between 290◦C and 500◦C. This

produces samples whose oxygen contents place them on the overdoped side of the phase

diagram, with sharp superconducting transitions ranging from 5 to 85K. We have not

attempted to reach optimal doping, which in this system is believed to correspond to

a superconducting Tc of at least 93 K [29], because even higher annealing temperatures

would be required, increasing the risk of thallium loss.

2.2. Physical and Chemical Analysis

Since cation substitution can cause increased scattering, one objective of this work was

to grow crystals with the lowest possible cation substitution level. Of more immediate

importance than the reduction of cation defects, however, is reproducibility. Because

cation substitution also dopes the crystal, variation between crystals would lead to

Tl2201 Brings Spectroscopic Probes Deep into the Overdoped Regime

9

Figure 4.

(a) The (0 0 10) rocking curve for a typical Tl2201 single crystal

(Tc=67.7K); the FWHM is 0.034◦, indicating excellent crystallinity. (b) The (6 0 0)

and (0 6 0) peaks for a Tc =24K sample, showing the orthorhombicity of the crystal

structure. The doublets correspond to CuKα1 and CuKα2 radiation results.

different Tcs as well as different scattering rates. More problematic still would be a

variation within a crystal, which would arise from changes in the composition of the

melt over the course of the crystal’s growth. Although cation substitution is not desired

and does dope the crystals, it must be emphasized that this is not used as a control

parameter in doping these crystals. If the crystals can be grown with a reproducible

level of cation substitution, the cation disorder will be the same for all dopings. This is

in contrast to LSCO, where cation substitution is the primary control parameter, and

must be varied by roughly a factor of four to traverse the superconducting dome.

To determine the levels of cation substitution in our crystals, electron-probe micro-

analyses (EPMA) of several crystals were performed on a fully-automated CAMECA

SX-50 instrument in wavelength-dispersion mode.‡ The cation composition (normalized

to the barium content) was determined to be Tl1.884(6)Ba2Cu1.11(1)O6+δ and was not

observed to be position dependent on individual crystals, nor did it vary between

crystals. This result corresponds to a level of copper substitution at the thallium site

comparable to that reported previously [49, 39, 40, 44] for cation-substituted Tl2201.

Al contamination was checked separately by EPMA and found to be below the 50 ppm

detection limit, corroborating our observation that the crucibles were not corroded by

the melt.§ Despite the cation non-stoichiometry, the high quality of these crystals is

‡ The EPMA was performed under the following operating conditions: excitation voltage, 15kV;

beam current, 20nA; peak count time, 80s; background count time, 40s; and spot diameter, 10 µm.

Data reduction was performed using the “PAP” ϕ(ρZ) method [48]. For the elements studied, the

following standards, X-ray lines and monochromator crystals were used: elemental Tl, TlMα, PET;

YBa2Cu3O6.920, BaLα, PET; and YBa2Cu3O6.920, CuKα, LIF. Tight Pulse Height Analysis (PHA)

control was used to eliminate to the degree possible any interference from higher-order lines.

§ Operating conditions were the following: excitation voltage, 15kV; beam current, 40nA; peak count

time, 500 s; background count time, 250 s; and spot diameter, 10 µm. The matrix composition was fixed

at stoichiometric; AlKα was standardized on α-Al2O3 using a TAP crystal monochromator.

Tl2201 Brings Spectroscopic Probes Deep into the Overdoped Regime

10

Figure 5. Schematic depiction of the dependence of the orthorhombic/tetragonal

structural transition on y and δ in Tl2−yBa2Cu1+yO6+δ. This scenario is suggested by

the data from polycrystalline samples of Wagner et al. [29] and our own single crystal

diffraction results (see Fig. 6). The CuO2 plane and unit cell (in green) are shown with

and without the orthorhombic distortion, which is here exaggerated for clarity. The

gray bar indicates where our crystals fall on this plot: y=0.116.

evidenced by their narrow superconducting transitions, measured by magnetization in

a Quantum Design SQUID magnetometer (MPMS). Fig.3(b) shows the field-cooled

magnetization measured at 1 Oe (H c), for crystals with transition widths (10%-90%)

ranging from 4 to 0.7K. The observed 40∼60% Meissner fraction (i.e., fraction of flux

excluded when cooling through Tc) compares well with field-cooled results on other

superconducting cuprates. It should also be mentioned that Tl2201’s extremely low

first critical field (Hc1) makes the transitions appear broader in higher fields.

2.3. Structural Analysis: Crystallinity and Symmetry

To investigate the samples’ crystallinity, X-ray diffraction spectra and rocking curves

were taken on a BEDE model 200 double-crystal diffractometer with a Si(111)

monochromator; the entire sample was illuminated by the X-ray beam. Fig.4(a)

depicts the (0 0 10) X-ray rocking curve of a 0.5×1.2 mm2 crystal annealed to

Tc=67.7K. While the FWHM of 0.034◦ is broader than the 0.006◦ obtained for the

best YBa2Cu3O7−δ crystals grown in BaZrO3 crucibles, it is similar to the 0.02-0.03◦

achieved for YBa2Cu3O7−δ grown in YSZ crucibles [50], indicating a good degree of

crystalline perfection. A crystal annealed to achieve a relatively high oxygen content

Tl2201 Brings Spectroscopic Probes Deep into the Overdoped Regime

11

Figure 6. Comparison of the orthorhombicity between our crystals and Wagner’s [29]

and Shimakawa’s [51] ceramics: orthorhombicity and c-axis lattice constant vs. Tc.

and strong overdoping (Tc=24 K) was found to be orthorhombic with lattice parameters

a=5.4580(3) Å, b=5.4848(5)Å and c=23.2014(5) Å, consistent with the lattice constants

reported for ceramics with similar Tc and Cu substitution [29]. This orthorhombicity is

determined by the x-ray diffraction data presented in Fig.4(b): the two data sets were

recorded independently for a and b axes and would be identical to one another if the

crystals were tetragonal. Closer to optimal doping, and at a lower interstitial oxygen

content, the Tc=67.7 K sample was tetragonal to within our resolution

Tl2201’s crystal symmetry depends on the level of cation substitution and the

oxygen content. Stoichiometric and near-stoichiometric ceramics have been shown to

be orthorhombic for all oxygen contents, while heavily-substituted ones are strictly

tetragonal [51]. For intermediate levels of substitution, the orthorhombicity depends on

the oxygen content: samples with higher oxygen contents are more orthorhombic [29].

The orthorhombic distortion is thought to arise from a lattice mismatch between the

CuO2 plane layer and the naturally larger Tl2O2 double layer. Interstitial oxygens

increase the distortion, while cation disorder seems to suppress it. The distortion

and its dependence on the oxygen content δ and the copper substitution y are shown

schematically in Fig. 5. Powder X-ray diffraction shows that the orthorhombic distortion

is an elongation along one plaquette diagonal, with the Cu-O bonds remaining the same

in both directions. The distortion is quite minor: instead of being at right angles,

the Cu-O bonds meet at ≈89.7◦ in orthorhombic samples. The plaquette diagonals,

identified with nodes in the superconducting gap, remain orthogonal, so no mixing of

order parameter symmetries is required. For these reasons and for ease of comparison

with other systems, all ARPES analysis reported here is based on the original tetragonal

unit cell, regardless of the actual symmetry of the crystals.

Our crystals’ orthorhombicity, defined as 2(b − a)/(b + a), is consistent with that

found by Wagner et al. [29] for similar levels of cation substitution, as shown in Fig. 6(a).

One should keep in mind, however, that this quantity is highly prone to uncertainties

Tl2201 Brings Spectroscopic Probes Deep into the Overdoped Regime

12

because of the smallness of the orthorhombic distortion, so that quantitative conclusions

based on this graph should be treated with caution. Fig.6(b) compares two of our

annealed crystals’ c-axis lattice parameters to the orthorhombic ceramic samples of

Wagner et al. [29] and Shimakawa et al. [51]. Our c-axis lattice parameters are again

similar to those of Wagner et al. [29], but we observe higher Tc values for the same

lattice parameters (or a shorter c axis for the same Tc). Possible explanations for this

include improvements in Tc from higher homogeneity, cleaner samples or larger grain

size (crystals vs. ceramics), more homogeneous annealing, or possibly a calibration

disagreement between the diffractometers. Wagner’s samples have transitions a few

Kelvins wide, with long tails towards zero Kelvin, and Tc was defined using the 50%

point, which may not be an accurate average of the bulk in such a situation.

3. The Low-Energy Electronic Structure of Tl2201

Angle-resolved photoemission spectroscopy (ARPES) is an important tool in the

study of the electron dynamics in correlated-electron systems. Within the sudden

approximation, it probes the energy and momentum dependence of the electronic

excitation spectrum of an N − 1 particle system, the so-called electron-removal portion

of the single-particle spectral function A(k,ω) [5]. In the non-interacting picture, this

spectral function consists of delta-function peaks located at the precise energy and

momentum given by the band structure, i.e. A(k,ω) = δ(ω−ǫk). When interactions

are considered, the single-particle spectral function is modified by the inclusion of the

electron proper self energy Σ(k,ω)=Σ′(k,ω)+iΣ′′(k,ω), which captures all the of the

many-body correlation effects. One can then write A(k,ω) = −1

π

Σ′′(k,ω)

[ω−ǫk−Σ′(k,ω)]2+[Σ′′(k,ω)]2 .

With respect to the non-interacting case, the peaks in the spectral function shift in

energy and gain a finite width, in a way dependent on the energy and momentum of the

excitations. At those ω and k for which the spectral function is still characterized by a

single pole, energy and lifetime renormalization are directly described by Σ′(k,ω) and

Σ′′(k,ω), respectively. The ARPES lineshape thus gives direct access to the lifetime of

the excitation and can provide insights into the nature of the underlying interactions, for

example whether or not electron-electron interactions are Fermi liquid-like, as discussed

in relation to the Tl2201 transport data of Fig. 2. As it will be discussed in the following

sections, the first successful ARPES experiments on Tl2201 have arrived at an agreement

with bulk probes on key features such as the normal-state Fermi surface [32] and the

superconducting gap [52]. This success suggests that detailed ARPES studies of Tl2201

have the potential to reveal the nature and strength of many-body correlations, upon

approaching high-Tc superconductivity from the more conventional overdoped regime.

3.1. Band Structure Calculations

Since it is generally believed that the normal metal on the very overdoped side of the

HTSC phase diagram can be described as a rather conventional Fermi liquid system, in

Tl2201 Brings Spectroscopic Probes Deep into the Overdoped Regime

13

Figure 7. (a) Electronic dispersion obtained from band structure calculations within

the local-density approximation (LDA). (b,c) Fermi surface calculated for two different

doping levels enclosing a volume, counting holes, of 50% (orange) and 63% (blue) of the

two-dimensional projected Brillouin zone; the corresponding position of the chemical

potential with respect to the dispersive electronic bands is indicated in panel (a). Here,

as in the rest of the paper, k-space labels are expressed modulo the lattice constants.

contrast to the strongly correlated Mott insulator found at half filling (p=0 in Fig.1),

exploring the electronic structure and Fermi surface of heavily overdoped Tl2201 can

start with the results of non-interacting band structure calculations performed within

the local density approximation (LDA). As shown in Fig. 2 and especially Fig. 5, the most

important structural element is the CuO2 planes, which are well separated from each

other by the TlO and BaO layers. As in all other HTSC cuprates, the CuO2-derived

bands are expected to be the lowest energy electronic states and therefore directly

determine the macroscopic electronic properties.

The results of band structure calculations along the high symmetry directions

in the body-centered tetragonal Brillouin zone of Tl2201 are presented in Fig.7(a).

Coordinates for the atoms are taken from Ref.[53]. The essential low-energy feature of

the band structure is indeed the highly dispersive Cu(3dx2−y2 )-O(1)(2px,y) band; note

however that this band is highly two-dimensional, with little dispersion along the kz

direction (0, 0,kz =0)→(0, 0,kz =π). Its top and bottom are located at (π, π) and (0, 0),

respectively, within the two-dimensional projected Brillouin zone, while at (π, 0) we find

Tl2201 Brings Spectroscopic Probes Deep into the Overdoped Regime

14

the well-known van Hove singularity. The other two bands that sit at a comparable

energy, i.e 0 to −2eV in Fig.7(a), are the anti-bonding Tl(6s)-O(2),O(3)(pz) bands,

which do show significant dispersion in the kz direction. Based on the formal valences of

stoichiometric and undoped Tl2Ba2CuO6+δ (2:2:1:6 with Tl3+, Ba2+, Cu2+, O2−, δ=0),

the Fermi energy would be located at the red line in Fig. 7(a), and both the CuO and TlO

bands would cross Ef . This would result in a Fermi surface with hole pockets associated

with the CuO band, and a small, spheroidal TlO electron pocket at the Γ = (0, 0)

point, as in Fig.7(b). This small electron pocket was originally proposed as a possible

explanation for why undoped Tl2Ba2CuO6 does not show the Mott insulating behavior

expected for materials with a half-filled 3dx2−y2 CuO band, such as undoped La2CuO4

[53]. However, the non-stoichiometry of our samples (Tl1.884(6)Ba2Cu1.11(1)O6+δ, see

Section 2.2) provides additional hole doping (∼ 0.14 holes/formula unit) that would

push the CuO band further away from half-filling, driving the TlO band above Ef . This

shift would generate a Fermi surface consisting solely of a CuO hole pocket around

(π, π) similar to the results shown in blue in Fig. 7(a,c), which were calculated to match

the 63% volume observed by ARPES on our Tc =30 K overdoped Tl2201 crystal [35].

Finally, it should be mentioned that if the effect of hole doping through interstitial

oxygen were explicitly included in the calculations, the TlO bands would be pushed to

much higher energies, beyond the rigid band picture [54]. The analogous effects of Pb

substitution or excess oxygen in the Bi-O layers of Bi-cuprates has recently been used

to account for the lack of Bi-O electron pockets around (π,0) in those materials [55].

3.2. Electronic Dispersion and Fermi Surface by ARPES

Fig. 8(b) shows ARPES data from a Tc =30 K very overdoped Tl2201 sample (Tl2201-

OD30) taken close to the (0,0)-(π,π) direction in momentum space, the so-called “nodal”

direction where the d-wave superconducting gap is zero. Each line in Fig.8(b) is an

energy distribution curve (EDC) from a given position in momentum space along the

line marked as I in the Fermi surface plot of Fig.8(a). These EDCs show a strongly

dispersing quasiparticle peak, related to the 3dx2−y2

CuO band, which emerges from

the background at high binding energies and progressively sharpens as it disperses all

the way to the Fermi energy EF . Note that the term quasiparticle (QP) is used in a

loose sense to identify a reasonably sharp dispersive peak, without specifying in detail

how the peak should be interpreted in terms of specific models for correlation effects

The ARPES experiments were carried out at the Swiss Light Source (SLS) on the Surface and

Interface Spectroscopy Beamline [56, 57, 58] and at the Stanford Synchrotron Radiation Laboratory

(SSRL) on Beamline 5-4, in both cases using a Scienta SES-2002 photoelectron spectrometer. At SLS,

all measurements were performed with circularly polarized 59eV photons, with energy and angular

resolutions of approximately 24 meV and 0.2◦; the data were acquired on a strongly overdoped sample

with Tc=30K (Tl2201-OD30), and a lightly overdoped sample with Tc=63K (Tl2201-OD63). The

samples were cleaved in situ at 6×10−11 torr and kept at 10 K throughout the measurements. At SSRL,

all data were acquired at 28 eV with linearly polarized light and with energy and angular resolutions of

15 meV and 0.35◦. The SSRL data were from lightly overdoped samples with Tc=74 K (Tl2201-OD74),

which were cleaved at 3×10−11 torr and temperature cycled between 10 and 85K.

Tl2201 Brings Spectroscopic Probes Deep into the Overdoped Regime

15

Figure 8. (a) Fermi surface calculated for a 63% Brillouin zone volume, counting

holes, as in Fig.7(c). (b,c) ARPES spectra taken at T =10K on Tl2201-OD30 along

the directions marked by arrows I and II in (a). (d) Second derivative vs. energy of

the spectra from along arrow III in (a); the red line is our tight-binding fit (see text).

(more discussion will be given in Sections 3.4 and 4). From the results of band structure

calculations (Fig. 7), we expect the bottom of this band to be located at the Γ point, or

(0,0), at a binding energy of approximately 1 eV with respect to the chemical potential.

Although it is hard to track the band all the way down to high binding energies in the

raw data, it is possible to highlight the band dispersion by taking a second derivative

of the ARPES intensity with respect to energy. This is plotted in Fig.8d and indicates

a filled bandwidth of only ∼ 250meV, suggesting a renormalization factor of about 4

between single particle band structure calculations and measured ARPES spectra.

Near (π,0), the so-called “antinodal” direction where the d-wave superconducting

gap exhibits its largest value, we also detected well-defined dispersive QP peaks, which

define a shallow parabolic band. The bottom of this band corresponds to the extended

van Hove singularity observed near (π,0) in all superconducting cuprates [2, 3]. However,

at this high doping level, the van Hove singularity is located approximately 40 meV below

the chemical potential. In comparison to the band structure calculations, this also shows

a renormalization factor of about 5.

The momentum at which a dispersing QP peak is observed to reach EF and

disappear provides ARPES’s means of determining a Fermi wavevector kF . This

wavevector identifies one point along the normal state Fermi surface, which is the

Tl2201 Brings Spectroscopic Probes Deep into the Overdoped Regime

16

Figure 9. Fermi surfaces, in the projected two-dimensional Brillouin zone, obtained

from: (a) band structure calculations (imposing a volume of 63% counting holes, as

observed by ARPES on Tl2201-OD30); our (b) ARPES experiments on the Tc = 30 K

sample (Tl2201-OD30); and (c) the AMRO study on a Tc = 20K Tl2201 crystal by

Hussey et al. [32]. The red line in (b) is the result of our tight-binding fit of ARPES

Fermi surface and dispersion; the width of the Fermi surface contour in (c) reflects the

magnitude of the c-axis dispersion, here emphasized by a factor of 4 for clarity [32].

contour in momentum space that separates filled from empty electronic states and

whose existence and details are of fundamental importance for the understanding of the

macroscopic physical properties of a material. By integrating the ARPES spectra over

a ±5 meV energy window about EF for a large number of cuts in momentum space and

then plotting the results versus momentum in the two-dimensional projected tetragonal

Brillouin zone, one obtains an estimate of the normal-state Fermi surface. This has been

done for Tl2201-OD30, and is shown in Fig.9(b). The location of the Fermi surface

crossings has been determined over more than one quadrant across two different zones,

and then downfolded to the first Brillouin zone and four-fold symmetrized to clearly

show the detailed shape of the Fermi surface in the reduced zone scheme, with improved

signal-to-noise. As expected, there is no indication of a TlO electron pocket around (0,0).

The Fermi surface determined from this procedure takes the form of a large hole pocket

centered at (π,π) with an area that occupies 63±2% of the Brillouin zone, corresponding

to a carrier concentration of 1.26±0.04 hole/Cu atom, or p=0.26±0.04 greater hole

density than the half-filled Mott insulator with 1 hole/Cu (p=0 in Fig.1). This result

is in superb agreement with the recent study of the Fermi surface by angular dependence

of magnetoresistance oscillations (AMRO) experiments [32], which found a hole-pocket

volume of 62% of the Brillouin zone in a slightly more overdoped Tc = 20K sample

(Fig.9(a)). The ARPES determination is also in good agreement with the estimates

from low temperature measurements of the Hall coefficient, which gave a hole doping

of p=1.30 holes/Cu for a Tc ≲15K sample [33]. It is worth noting that the commonly

used empirical formula, Tc/Tmax

c

= 1−82.6(p−0.16)2, which purports to connect the

value of Tc to doping in a universal fashion [59], gives a much stronger dependence of Tc

on doping than is shown by ARPES, AMRO, and Hall coefficient data. For our Tl2201-

OD30 sample, from the above formula and the value p=0.26 given by the Fermi surface

Tl2201 Brings Spectroscopic Probes Deep into the Overdoped Regime

17

volume, one would obtain Tc ≃16 K; and for the Tc =15 K sample studied by Mackenzie

et al. [33], with the formula and the value p=0.30 determined from the Hall coefficient,

one would actually get Tc = 0K. According to the empirical formula and contrary to

what observed, Tl2201 should become non-superconducting already for p≃0.27.

As a quantitative measure of the shape of the Fermi surface and of the many-body

renormalized electronic dispersion, the Tl2201-OD30 ARPES data can be modelled

by the tight-binding formula ǫk = µ+ t1

2

(cos kx + cosky)+t2 coskx cosky + t3

2

(cos 2kx +

cos 2ky)+ t4

2

(cos 2kx cos ky +coskx cos 2ky)+t5 cos 2kx cos 2ky [60], setting a = 1 for the

lattice constant. With parameters µ = 0.2438, t1 = −0.725, t2 = 0.302, t3 = 0.0159,

t4 =−0.0805, t5 =0.0034, all expressed in eV, this dispersion reproduces both the Fermi

surface shape, solid red line in Fig.9(b), and the QP energy at (0,0) and (π,0), as

seen in Fig.8(c,d). It is worth noting that experimentally the band bottom at (π,0)

is extremely flat, a behavior that could not be reproduced by including only t1 and t2

hopping parameters in the model. Alternatively, a simple analytical formula for the

three-dimensional electronic dispersion of Tl2201 has recently been derived, within the

framework of the linear combination of atomic orbitals (LCAO) approximation, and was

used to fit both ARPES and AMRO results [61]. A basis set spanning the Cu 4s, Cu

3dx2−y2 , O 2px and O 2py states was used in order to take into account the effective

Cu-Cu hopping amplitude between Cu 4s orbitals in neighboring CuO2 layers. This

approach emphasizes the significance of the Cu 4s hopping amplitude in order to obtain

the correct three-dimensional Fermi surface shape, an issue that was originally raised

by Andersen et al. [62]. With regards to the detailed shape of the Fermi surface, LDA

band structure calculations predict a more square contour than observed by ARPES and

AMRO (Fig.9(a)). The inclusion of correlation effects, as well as Cu-Tl substitution

or interstitial O-doping beyond a rigid-band picture, might lead to a better agreement;

preliminary attempts in this direction have not yet led to qualitative improvements [54].

3.3. ARPES Study of the Superconducting Gap

In order for Tl2201 to be a model system to study the overdoped HTSCs by surface and

bulk sensitive probes, it is important to show that ARPES measures quantities that are

characteristic of the bulk, for both normal and superconducting states. Therefore, in

addition to the quantitative agreement on the normal state Fermi surface, the observance

of a superconducting gap in agreement with bulk measurements is also a necessary

requirement. In the following, we will discuss three different methods for the observation

of a gap by ARPES. The first two show a gap consistent with a dx2−y2 form. The third

method allows one to follow the temperature dependence of the gap, highlighting the

minimal surface degradation that occurs as the temperature is cycled from 10 to 85 K.

The detection of a dx2−y2

gap using ARPES can be most easily visualized by

the comparison of nodal and antinodal symmetrized spectra [64]. The spectra are

symmetrized in energy about Ef , by taking I(ω)+I(−ω), which minimizes the effects

Tl2201 Brings Spectroscopic Probes Deep into the Overdoped Regime

18

Figure 10. (a,b) Symmetrization of Tl2201-OD30 ARPES spectra from along cuts I

and II in Fig.8(b,c). ARPES spectra and their symmetrization along (c,d) the nodal

direction for overdoped Bi2Sr2CaCu2O8+δ (Tc =88K), and (e,f) perpendicular to the

antinodal direction for overdoped Bi2Sr2CuO6+δ (Tc =23K); after Ref.[63]. For all

samples, the approximate locations of the k-space cuts are indicated in the Brillouin

zone sketches. Bold lines in (a,c,d) mark the spectra that cross the Fermi energy,

identifying a Fermi wavevector kF ; no crossing is observed for the spectra in (b,e,f).

of the Fermi function.¶ While this procedure does not return a quantitative value for

the size of the superconducting gap, it provides a qualitative criterion for determining

whether or not there is a Fermi crossing, and hence whether or not a superconducting

gap has opened along the normal-state Fermi surface. In the symmetrized ARPES

data, the presence of a peak in the spectra at EF indicates the presence of a Fermi

surface crossing. This procedure has been used extensively for the Bi-based cuprates,

both single and bilayer compounds, in detailed investigations of the normal-state Fermi

surface, as shown for instance in Fig. 10(c-f) [63], and of the superconducting as well as

normal state pseudogap [64, 66]. In the case of our Tl2201-OD30 sample, this crossing

is clearly seen in the nodal direction (bold line in Fig.10(a)) but not in the antinodal

direction (Fig.10(b)), which is consistent with a d-wave functional form for the gap.

A more quantitative analysis of the gap can be performed by fitting ARPES spectra

along cuts that cross the underlying normal-state Fermi surface, as shown in Fig. 11. The

¶ The symmetrization procedure assumes that the ARPES spectra are described by I(ω)=f(ω)A(k,ω),

and that there is particle-hole symmetry for a small range of ω about EF such that A(−ǫk, −ω) =

A(ǫk,ω). With the identity f(−ω)=1−f(ω), it then follows that I(ω)+I(−ω)= A(k,ω). It is worth

pointing out that this procedure is not strictly valid in the case that energy and momentum resolutions

are included in the description of I(ω), via convolution with the resolution function R(∆k, ∆ω).

Tl2201 Brings Spectroscopic Probes Deep into the Overdoped Regime

19

Figure 11. (a) Enlarged view of the Fermi surface of Tl2201-OD30 near (π,0). (b)

Selected spectra from along cut II in (a); their k-space positions are indicated by circles

of corresponding color. (c,d) QP linewidth Γ and peak position from a Lorentzian fit of

the energy distribution curves along cut II in (a). (e) Similarly, QP peak position along

cut III in (a). Black lines in (a,d,e) are our tight-binding results (see text). (f) Normal

and (g) superconducting state single-particle spectral function, highlighting particle-

hole mixing and backward dispersion of Bogoliubov QPs below Tc (after Ref. [65]).

model used to reproduce the spectra is a Lorentzian peak plus a step-like background.

The latter is determined from the ARPES spectra with k ≫ kF and is used to help

phenomenologically isolate the coherent part of the spectral function. This function

is then multiplied by the Fermi function and convolved with the instrumental energy

resolution function, to obtain the functional form to be fit to the data [2]. This procedure

for determining the gap has been used previously for Bi2Sr2CaCu2O8+δ [67]. The inset

of Fig.11(b) shows good agreement between the raw data and the fit. Since heavily

overdoped cuprates have weaker, possibly Fermi liquid-like electron correlations, good

agreement between the measured QP peak and the Lorentzian is in principle expected.

Fig. 11(d,e) show the peak positions from the fits compared with our tight-binding

description of the normal-state electronic structure near the antinodal region. At higher

binding energies, there is good agreement between the fit peak positions and the normal-

state dispersion. At lower binding energies, however, the peak does not reach EF , but

instead reaches a minimum at ∆P ≃17meV and then disperses back to higher binding

energy. This behavior is a hallmark of Bogoliubov QPs in a superconductor, as shown in

the sketch of Fig. 11(f,g), for which a beautiful experimental demonstration was recently

obtained by ARPES on the trilayer Bi-cuprate Bi2Sr2Ca2Cu3O10+δ [65]. Simultaneously,

Tl2201 Brings Spectroscopic Probes Deep into the Overdoped Regime

20

Figure 12.

Symmetrized ARPES spectra measured at (a) 10K and (b) 85K on

overdoped Tl2201-OD74, along the (π, 0)−(π, π) direction. Bold lines indicate the

EDCs closest to kF . (c) The k integrated spectral weight at the antinode, for several

temperatures, together with a 10K Au Fermi edge and an 85K resolution-broadened

Fermi function; an enlarged view of the leading edge positions is given in the inset.

we observe a reduction of spectral weight, by about a factor of 2, when the QP peak

has dispersed from the van Hove singularity all the way to kf . This is consistent with

the spectral intensity of Bogoliubov QPs being determined by the coherence factor

v2

k

=1−u2

k

= 1

2

(1−ǫk/Ek), with Ek =√ǫ2

k +∆2

k

, which corresponds to 1/2 when ǫk =ǫkf =0

and 1 for |ǫk|≫∆k. Extending this analysis to other momenta along the Fermi surface,

one can study the k-dependence of the gap: at ∼(π/2, π/2) the peak does cross EF (not

shown), while at intermediate momenta a gap smaller than at (π, 0) is observed, as on

the right-hand side of Fig.11(d), consistent with d-wave symmetry.

The temperature dependence of the gap at the antinode has been measured for

Tl2201-OD74. Fig.12(a,b) present symmetrized ARPES data along the (π, 0)−(π, π)

direction, at temperatures above and below Tc. In exactly the same manner as the

k-dependent symmetrized data of Fig.10(a,b), the temperature-dependent data of

Fig.12(a,b) show a clear gap at k = kF at 10K, which is clearly smaller at 85K (with

this type of analysis, the broadness of the QP peak and noise level at 85 K make it hard

to conclude whether the gap has completely closed). Fig.12(c) shows averaged EDCs

taken over a narrow k-range around kf at 10, 45, 85, and again at 10K. Included in

the figure is a 10K Au reference spectrum, and the simulation of an energy-resolution

broadened Fermi function at 85K. The inset of Fig.12(c) shows a closer view of these

k-integrated EDCs around Ef . It is clear from this figure that, as the temperature

increases, the size of the gap decreases although it does not seem to completely close

at 85K. This is suggested by comparing the position of the leading edge of the 85K

ARPES data with the simulated Fermi function; they are close in energy but not quite

coincident. It is possible that there is still a pseudogap at temperatures slightly above

Tl2201 Brings Spectroscopic Probes Deep into the Overdoped Regime

21

Figure 13. (a) Superconducting gap magnitude estimated by tunneling and ARPES

from the superconducting peak position (SCP) and the leading-edge midpoint (LEM)

shift, plotted versus Tc for various optimally or overdoped cuprates (after Ref. [68]). (b)

Gap values from thermal conductivity, tunneling, and ARPES plotted vs. |1−Tc/Tmax

c

|

for many HTSCs across the phase diagram (after Ref. [52]); dashed lines are guides to

the eye. Both plots include our own Tl2201-OD74 and Tl2201-OD30 ARPES data.

Tc = 74K at this doping level. It is worth emphasizing that as the temperature is

lowered back to 10K, the gap reopens to approximately the same size. Although a

partial degradation of the ARPES features can be seen, the QP peaks and the gap are

still clearly observable. This is strong evidence that the surface is stable under these

experimental conditions and is thus suitable for detailed surface-sensitive experiments.

From these data, a superconducting peak (SCP) position of ∼33 meV and a leading-edge

midpoint (LEM) gap of ∼15 meV can be extracted.

As a direct comparison with other cuprates and different means of determining

the gap, Fig.13(a,b) present a compilation of the doping dependence of the gap

magnitude. Fig.13(a) refers only to spectroscopic studies of optimally and overdoped

HTSCs, for which the gap is proportional to Tc. Our results from Tl2201-OD74 and

Tl2201-OD30 are fully consistent with the behavior observed on the other cuprates,

as far as both SCP and LEM positions are concerned. Even more significant is

Fig.13(b), which demonstrates that ARPES measures gap values that follow the same

trend as those derived from thermal conductivity, a bulk transport property. A more

accurate comparison would require an analysis of the ARPES data beyond the mere

determination of SCP and LEM positions, for instance on the basis of a model spectral

function; this is however complicated by the non-trivial doping dependence of the

ARPES data, and in the present case was done for Tl2201-OD30 but not Tl2201-OD74.

Tl2201 Brings Spectroscopic Probes Deep into the Overdoped Regime

22

3.4. ARPES Lineshape Analysis

The electrical and thermal transport properties suggest that overdoped Tl2201 may be

regarded as a Fermi liquid [33, 34, 32]. Since the ARPES data for both the normal and

superconducting states indicate that single-particle surface-sensitive probes do provide

information representative of the bulk, overdoped Tl2201 might be an ideal material

for a quantitative study of the strength and nature of the many-body effects, which are

revealed by the energy and momentum evolution of a QP’s intrinsic ARPES lineshape

[5]. In general, however, the shape of an ARPES peak does not correspond directly to

an excitation’s intrinsic lineshape. This is primarily due to an extrinsic contribution

from momentum and energy resolution of the analyzer. Recent work on the model

two-dimensional Fermi liquid system Sr2RuO4 highlights the significant issues involved

in removing the effects of the analyzer resolution [69]; in particular, it clearly shows

that both the lineshape width and the peak position can be affected by resolution

effects. There are significant challenges to completely removing the analyzer resolution

from the data and stringent requirements on the data quality needed to attempt such

analysis; therefore, we will analyze the current Tl2201 spectra in a more qualitative

and phenomenological manner, concentrating primarily on the momentum dependence

of the QP peaks measured in the superconducting state for different doping levels.

Beyond the issue of the experimental resolution, matrix element effects and the

handling of the ARPES background are the two main complications in attempting

even a qualitative description of the momentum dependence of the QP lineshapes.

Experimentally, problems due to matrix element effects may be reduced by appropriate

choice of measurement conditions such as geometry, photon energy, and polarization.

For the experiments performed at the SLS and the data discussed in the following,

circularly polarized 59eV photons were used. Circular polarization was chosen

specifically to minimize matrix element effects, which, using symmetry arguments, can

be shown to be most extreme for linear polarization [2]. The photon energy was selected

based on our experience on other cuprates — specifically, La2−xSrxCuO4 (LSCO) where

similar photon energies were used to reveal structure that had previously been missed in

lower-energy ARPES data [70, 26]. Finally, in order to gain more reliable information,

the ARPES data were taken over multiple Brillouin zones; this way it was possible to

verify that, for the same point in the reduced zone scheme, the lineshapes from different

zones in the extended zone scheme were qualitatively the same.

The most commonly used method to isolate a QP peak from the experimental

background [2, 71], is to subtract a phenomenological ARPES background determined

from the data at k-values far removed from those at which the QP-like peaks are

detected. In our analysis we will use this approach, taking as background the weakly

k-dependent photoemission intensity observed in each momentum space cut for k ≫ kF .

The fundamental problem with this approach is that if some of the background is related

to the incoherent part of the spectral function, as one would expect for correlated

electron system, then by subtracting the background we are actually disregarding some

Tl2201 Brings Spectroscopic Probes Deep into the Overdoped Regime

23

Figure 14. (a) Tl2201-OD30 ARPES spectra at k slightly smaller than kF along

the Fermi surface contour (corresponding to a QP binding energy of ∼35meV). (b)

Selected spectra from (a) along with their corresponding k ≫ kF background. (c)

Spectral weight of the background subtracted spectra, integrated over different energy

ranges and normalized with respect to the α=0 antinodal value, plotted vs. the Fermi-

surface angle α. (d) Quasiparticle linewidth Γ plotted vs. the Fermi-surface angle α.

of the most crucial information contained in the ARPES spectra. In fact, the k- and

ω-dependence of the incoherent part of the spectral function is, in many respects, as

important as the behavior of the QP peak itself; unfortunately we do not yet have the

means to reliably extract information from it.

In Fig. 8(b), similar to what was previously observed in all photoemission studies of

the cuprates [2, 3], we saw that in the nodal region the width of the QP peaks increases

as a function of binding energy, as expected from simple phase-space arguments. This

is, however, in sharp contrast to what is observed in the antinodal region, where the

sharpest peak is found at the bottom of the band: in Fig.11(c), the linewidth Γ of the

QP peak is observed to grow from ∼30 to ∼55 meV as the QP peak disperses from ∼39 to

∼20 meV binding energy. Possible origins for this anomalous behavior will be discussed

later, but it should be pointed out that an analogous effect was recently reported for

very overdoped and nearly non-superconducting Bi1.74Pb0.38Sr1.88CuO6+δ [72]. Another

striking feature of the lineshape evolution is that the QP peaks are much broader in

the nodal region than in the antinodal region, as evidenced by the direct comparison of

Fig. 8(b) and 8(c). To elaborate on this, in Fig. 14(a) we present a compilation of spectra

from along the Fermi surface contour, with k slightly smaller than kF and corresponding

to a binding energy of ∼35 meV. The choice in favor of spectral peaks at a binding energy

Tl2201 Brings Spectroscopic Probes Deep into the Overdoped Regime

24

Figure 15.

Nodal and antinodal spectra from (a) Tl2201-OD63 and (b) Tl2201-

OD30, at k ≲ kF . Inset: similar data from overdoped LSCO-OD24 (p = 0.22), from

Ref. [73]. For both samples in (a), Tc ≃ 2

3 Tmax

c

with respect to the Tmax

c

of each family

[27].

lower than EF is dictated by the need to compare spectra not affected by either the

opening of the d-wave gap or the anomalous low-binding-energy broadening seen at

the antinode. The sharp peak near (π, 0) becomes progressively broader as (π/2, π/2)

is approached. In order to determine whether this apparent broadening represents an

increase in the QP linewidth Γ, or is merely a loss of spectral weight, the momentum-

independent background from k ≫ kF is subtracted from the ARPES spectra as shown

in Fig.14(b); then the data are integrated over varying energy ranges and plotted in

Fig. 14(c), normalized at the Fermi surface angle α=0◦. For narrow integration windows,

there is a drop as a function of α: the QPs show a loss of low-energy weight (and possibly

coherence) in the nodal region. However, once the integration window is expanded to

about 550meV, the spectral weight of the QP peaks becomes angle-independent. This

indicates that the k-dependent broadening of the ARPES spectra indeed reflects a loss

of coherence of the QP spectrum, rather than being due to matrix element effects.

The observed QP anisotropy of Tl2201-OD30, with peaks much broader in the

nodal than the antinodal region, is in sharp contrast to the behavior observed in

underdoped cuprates, where the QP peaks are sharp near (π/2, π/2) and very broad

around (π, 0), as summarized in Fig.17. As doping is increased, the antinodal QP

peaks sharpen but remain significantly broader than the nodal QPs up to optimal

doping, at least in the normal state. Even in the superconducting state, where antinodal

QPs sharpen considerably, the ARPES linewidths for underdoped and optimally-doped

materials are still highly anisotropic, with a minimum at (π/2, π/2) [2, 3]. Overall

this behavior leads to the expectation that increasing doping beyond the optimal value

will simply render the QP linewidths progressively more isotropic. Indeed, isotropic

lineshapes were observed on early data from very overdoped, non-superconducting

Tl2201 Brings Spectroscopic Probes Deep into the Overdoped Regime

25

Figure 16.

(a) ARPES intensity map for emission from the Fermi energy for

overdoped LSCO-OD24 (p=0.22). Red lines are the calculated LDA three-dimensional

Fermi surface for kz =0 and π/c; the enclosed area is the projection of the Fermi surface

onto the two-dimensional kx-ky plane, and denotes the region allowed for emission [74].

(b) ARPES and AMRO data from Tl2201-OD30. In (a,b) the yellow arrows identify

directions characterized by zero kz dispersion as indicated by the LDA calculations

and consistent with AMRO experiments, which also suggest much weaker residual

three-dimensionality effects in Tl2201 than in LSCO. (c) Resolution contributions to

the width of an ARPES EDC calculated, as a function of the Fermi velocity vF , from

the two-dimensional convolution of a Lorentzian quasiparticle peak (binding energy

ω = 30meV and intrinsic width Γ = 18meV) with Gaussian energy and momentum

resolution functions (results corresponding to our experimental parameters are in red).

Bi1.80Pb0.38Sr2.01CuO6−δ [75], although more recent work on the same material [72]

exhibits a behavior more consistent with what is reported here for Tl2201. Going back

to the QP anisotropy reversal observed for Tl2201-OD30 in Fig. 14, and presented again

in Fig. 15(b) for nodal and antinodal points only, it must be emphasized that the same,

albeit less pronounced, effect was observed in our Tl2201-OD63 sample, as shown in

Fig. 15(a). For comparison, in the inset of Fig. 15(a) we have plotted ARPES data from

overdoped (x=p=0.22) La2−xSrxCuO4 [73]. This sample has Tc ≃ 2

3

Tmax

c

, comparable

to the degree of overdoping of our sample Tl2201-OD63, which also has Tc ≃ 2

3

Tmax

c

for the much larger Tmax

c

of the Tl2201 family. The data from LSCO exhibit a QP

anisotropy similar to that in Tl2201-OD63, suggesting that the observed QP anisotropy

reversal might indeed be generic to the overdoped cuprates.

Before proceeding to the discussion of the broader significance of these findings, we

comment here on two extrinsic effects that could potentially contribute to an anomalous

k-dependent broadening of the QP lineshapes: residual kz electronic dispersion and

resolution broadening effects. First, the dispersion of the electronic structure along the

c-axis may indeed give rise to k||-dependent broadening of the ARPES features [74]

Tl2201 Brings Spectroscopic Probes Deep into the Overdoped Regime

26

(where k|| is the ab-plane momentum). This effect could be doping-dependent because

there is a 0.4% c-axis lattice constant reduction as doping is increased [29]. However, the

kz-dispersion vanishes in Tl2201 along the (0, 0)−(π, π) and (π, 0)−(π, π) directions, and

thus at all nodal and antinodal points. This is a direct consequence of the symmetry of

the body-centered tetragonal unit cell, and holds also for LSCO. Fig.16(a,b) illustrate

this point by showing ARPES data for both materials together with band structure

calculations for LSCO [74], and AMRO data for Tl2201 [32]. Along these high-symmetry

directions, no kz-dispersion is present either in the LSCO calculations or the Tl2201

AMRO data (in both cases the kz-dispersion is proportional to the width of the three to

two-dimension projected contours). This is consistent with the ARPES maps for the two

systems, although the broader EF intensity patches for LSCO indicate that the three-

dimensionality is much stronger in LSCO than in Tl2201. A residual kz dispersion might

contribute to the overall width of the ARPES features, but it cannot be responsible for

the reversal of the anisotropy observed on overdoped cuprates, since the anisotropy

exhibits a monotonic k||-dependence between (π/2, π/2) and ∼(π, 0).

The second extrinsic effect that influences the width in energy of an ARPES

spectrum, or EDC, is instrumental resolution. In a two-dimensional system, the width

in energy is determined by the intrinsic inverse lifetime of the QP excitation and by the

experimental resolution broadening, which reflects both the energy (∆ω) and angular

(∆k) resolution of the apparatus [5]. For a dispersionless feature, the instrumental

contribution to the total EDC width is determined solely by the energy resolution ∆ω.

However, for a dispersive feature, the angular resolution ∆k contributes to the energy

broadening in a manner directly proportional to the band velocity (vF =0 in Fig. 16(c)).

Since the Fermi velocity around the Fermi surface of the cuprates is highly anisotropic,

varying from approximately 0.5 to 1.8 eVÅ in going from antinodal to nodal region, this

should give rise to a momentum-dependent resolution broadening of the EDCs. This

broadening would be more severe at the nodes, where vF is at its largest, which raises

the concern that the larger nodal widths shown in Fig. 14 and 15 might be an artifact of

the instrumental resolution. We have carefully evaluated the contribution of resolution

broadening effects for our experimental conditions with a variety of different procedures

and the results are summarized in Fig.16(b). For our experimental parameters (red

data, ∆ω = 24meV and ∆k = 0.3◦) the k-dependent resolution broadening is of the

order of few meV, nowhere near the factor of 4 observed in Fig.14(d).

4. Discussion and Conclusions

Using a self-flux method and partial encapsulation, we have grown single crystals

of the single-layer overdoped cuprate superconductor Tl2201. The crystals were

annealed under controlled oxygen partial pressures to set a homogeneous doping level

whilst preventing decomposition, and typically exhibited sub-Kelvin superconducting

transition widths as measured in a SQUID magnetometer. Their high quality and

homogeneity were evidenced not only by these narrow transition widths but also by

Tl2201 Brings Spectroscopic Probes Deep into the Overdoped Regime

27

Figure 17.

Nodal-antinodal quasiparticle anisotropy reversal observed in the

cuprates. As a function of the Fermi surface angle α, sharp QP peaks are observed at

the nodes (N) and ill-defined QPs at the antinodes (A) in underdoped cuprates (e.g.,

data from Ca2−xNaxCuO2Cl2, after Ref.[10]); in overdoped Tl2201, this behavior is

reversed.

their extremely narrow rocking curve widths, comparable to YBa2Cu3O7−δ grown in

YSZ crucibles [50]. An EPMA study found the crystals to have the composition

Tl1.884(6)Ba2Cu1.11(1)O6+δ, which was homogeneous and was not observed to vary

between crystals. This level of cation substitution is similar to that reported earlier for

single crystals of Tl2201. For higher dopings (lower Tcs), the crystals were found to be

orthorhombic, a slight deviation from a square-planar lattice that is often associated with

the unsubstituted phase which has only been prepared in ceramics. The orthorhombic

distortion appears to be suppressed by cation disorder, but stabilized by the oxygen

interstitials that do increase the doping into the very overdoped regime.

While these crystals constitute a significant step forward, much optimization of

the growth technique remains, and there is still the challenge of growing single crystals

without the cation disorder in the Tl layer. Still, the high quality Tl2201 single crystals

grown as part of this effort have enabled the first successful ARPES experiments of

this compound. The results on the Fermi surface of Tl2201 mark an important new

starting point for understanding the cuprates, namely that there is a material where both

a surface-sensitive single-particle spectroscopic technique (ARPES) and comparable

bulk transport measurements (AMRO) have arrived at quantitative agreement on a

major feature of the normal state [32, 35]. This is a first for the copper-oxide high-Tc

superconductors and, within the more general class of 3d and 4d transition metal oxides,

it is second only to the case of Sr2RuO4 for which a similar quantitative agreement was

reached between de Haas-van Alphen [76, 77] and ARPES [69, 78, 79, 80, 81]. The

detailed agreement on the Fermi surface (Fig.9), together with the good quantitative

agreement achieved for the superconducting gap by thermal conductivity and ARPES

(Fig.13), establishes Tl2201 as an ideal system to study the overdoped regime of the

Tl2201 Brings Spectroscopic Probes Deep into the Overdoped Regime

28

cuprate phase diagram with a wide spectrum of techniques. Next it will be important

to study the single-particle excitation spectrum with both ARPES and STS in enough

detail to compare to other normal state properties, thus critically testing whether or not

a conventional Fermi-liquid description of the electronic properties captures the physics

of the normal metal in the heavily overdoped regime. In this regard, it is important

to elaborate on the most surprising result that emerged from this study of overdoped

Tl2201 by ARPES, one which is counterintuitive within the realm of Fermi liquid theory:

the quasiparticle anisotropy reversal observed across optimal doping. A summary of the

momentum anisotropy in under and overdoped cuprates is given in Fig.17, where data

from Ca2−xNaxCuO2Cl2 [10], for several doping levels, and Tl2201 are directly compared.

Early magnetotransport experiments on very overdoped Tl2201, for which a small

magnetoresistance and a weak T-dependence of the resistivity and cotangent of the Hall

angle have been observed, do not support the presence of strongly k-dependent low-

temperature scattering rates in the normal state [33]. More recently, a temperature,

doping, and magnetic field dependent AMRO study of Tl2201 suggested a two-

component scattering rate consisting of an isotropic T2 term as well as an anisotropic

T-linear term [82], which vanishes at the nodes and has a maximum near the antinodes.

The anisotropic term becomes weaker upon overdoping the material, which is consistent

with the view that as hole doping increases, the antinodal QPs should gain coherence

faster than the nodal ones, eventually leading to a relatively small and fully isotropic

scattering rate. Therefore, what is at variance between the ARPES and AMRO results

from Tl2201 is not the behavior of the antinodal QPs, which in both cases are indeed

gaining coherence upon overdoping, but rather that of nodal QPs which in ARPES

become progressively less coherent. In addition, the overall magnitude of the scattering

rate seen by ARPES is approximately a factor of 10 larger than estimated by AMRO.

What could be the origin for this discrepancy between ARPES and AMRO

determined scattering rates? First of all, one has to note that while the ARPES

experiments are performed in the superconducting state, the AMRO measurements

are carried out in the normal state; and the latter is achieved by the application of

external magnetic fields as high as 45T, which might potentially affect the low-energy

QP dynamics. As for the reliability of the ARPES data on this specific point of the loss of

coherence of nodal QPs at large dopings, it is important to mention that these findings

are also supported by a preliminary study of overdoped Bi2Sr2CaCu2O8+δ by STM

and STS [83]. From the Fourier analysis of the energy-dependent spatial modulations

observed in the tunneling conductance [84], it was concluded that for strongly overdoped

samples, the nodal (i.e., low energy) quasiparticle interference signal is no longer visible,

consistent with a decoherence of the nodal states [83].

A reconciliation of the results might require consideration of the specific

sensitivity of transport probes and single-particle spectroscopies to electronic scattering

phenomena. In particular, photoemission and scanning tunneling are very sensitive

to electronic scattering involving small momentum transfer, while transport is rather

insensitive to it. A possible source of such scattering could be extended impurities

Tl2201 Brings Spectroscopic Probes Deep into the Overdoped Regime

29

far from the CuO2 planes [85], such as cation substitution or interstitial oxygen which

increase with doping in most cuprates. Recent calculations show that the inclusion of

elastic forward scattering in a d-wave superconductor can lead to an overall increase of

the scattering rate in the normal state, but also to a strong enhancement of antinodal

QP coherence in the superconducting state [86]. A very different model, based on the

description of unitary-limit scattering beyond the Born approximation so as to account

for multiple scattering from a single impurity, would also lead, below Tc, to loss of

nodal coherence and anomalous enhancement of antinodal coherence [87]. However,

both models predict an approximately isotropic QP scattering above Tc, as well as

temperature dependent effects limited to the energy scale of the gap itself. To test the

applicability of these scenarios to the present case, a thorough temperature dependent

study is required. Finally, it should also be pointed out that sources of scattering relevant

to this discussion are not just limited to impurities; also inelastic (i.e., electronic)

scattering involving small momentum transfer might result in anomalous decoherence

of the nodal QPs. For instance, low-energy (i.e., small q) quantum-critical fluctuations

associated with proximity to a competing superconducting dx2−y2 +idxy phase [88], or

ferromagnetic phases, could also lead to a similar QP anisotropy reversal [89].

Acknowledgments

We gratefully acknowledge D.G. Hawthorn, K.M. Shen, N.E. Hussey, A.P. Mackenzie,

J.C. Davis, D.J. Scalapino, and G.A. Sawatzky for discussions and M. Platé, N.P.

Armitage, A.B. Kuzmenko, S. Chiuzbaian, M. Shi, M. Falub, and L. Patthey for

assistance during the ARPES experiments. We are also grateful to machine and

beamline groups at the Swiss Light Source and Stanford Synchrotron Radiation

Laboratory, whose outstanding efforts made the experiments possible. This work

was supported by the Canada Research Chairs Program, the Natural Sciences and

Engineering Research Council of Canada, the Canadian Institute for Advanced

Research, and the British Columbia Synchrotron Institute.

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