of 12
1
Scanning Tunneling Spectro
scopic Studies of the
Low-Energy Quasiparticle Excitations in Cuprate
Superconductors
N.-C. Yeh, M. L. Teague, R. T.-P. Wu, Z. J. Feng, H. Chu, A. M. Moehle
Department of Physics, California Institute of Technology, Pasadena, CA 91125
1-626-395 4313
1-626-395 3955
ncyeh@caltech.edu
http://www.its.caltech.edu/~yehgroup
Original paper for the Proceedings of ICSM2012, Istanbul (April 29 – May 4)
We report scanning tunneling spectroscopic (STS) studies of the low-energy quasiparticle
excitations of cuprate superconductors as a function of magnetic field and doping level. Our
studies suggest that the origin of the pseudogap (PG) is associated with competing orders (COs),
and that the occurrence (absence) of PG above the superconducting (SC) transition
T
c
is associated
with a CO energy
CO
larger (smaller) than the SC gap
SC
. Moreover, the spatial homogeneity of
SC
and
CO
depends on the type of disorder in different cuprates: For optimally and under-doped
YBa
2
Cu
3
O
7

(Y-123), we find that
SC
<
CO
and that both
SC
and
CO
exhibit long-range spatial
homogeneity, in contrast to the highly inhomogeneous STS in Bi
2
Sr
2
CaCu
2
O
8+x
(Bi-2212). We
attribute this contrast to the stoichiometric cations and ordered apical oxygen in Y-123, which
differs from the non-stoichiometric Bi-to-Sr ratio in Bi-2212 with disordered Sr and apical oxygen
in the SrO planes. For Ca-doped Y-123, the substitution of Y by Ca contributes to excess holes
and disorder in the CuO
2
planes, giving rise to increasing inhomogeneity, decreasing
SC
and
CO
,
and a suppressed vortex-solid phase. For electron-type cuprate Sr
0.9
La
0.1
CuO
2
(La-112), the
homogeneous
SC
and
CO
distributions may be attributed to stoichiometric cations and the
absence of apical oxygen, with
CO
<
SC
revealed only inside the vortex cores. Finally, the vortex
core radius (
halo
) in electron-type cuprates is comparable to the SC coherence length
SC
, whereas
halo
~ 10
SC
in hole-type cuprates, suggesting that
halo
may be correlated with the CO strength.
The vortex-state irreversibility line in the magnetic field versus temperature phase diagram also
reveals doping dependence, indicating the relevance of competing orders to vortex pinning.
Keywards: cuprate superconductiv
ity; pseudogap; competing orders;
quasiparticle excitat
ions; scanning tunneling spectroscopy
Abbreviations: SC; PG; COs; STS; LDOS; FT-LDOS; Y-123; Bi-2212, La-112
2
1. Introduction
The low-energy quasiparticle excitati
ons of cuprate superconductors exhibit
various spectral characteristics that
differ from those of simple Bogoliubov
quasiparticles for pure superconductors because of the presence of competing
orders (COs) in the ground state of unde
r- and optimally doped cuprates [1-10].
Some of the best known unconventional spectral characteristics include: the
presence (absence) of pseudogap and Fermi arc phenomena in hole-type (electron-
type) cuprates [4-12]; dichotomy of th
e quasiparticle coherence for momentum
near the nodal and anti-nodal parts of the Fermi surface [9,12,13]; pseudogap
(PG)-like spectral features inside vortex
cores [4-7]; and non-universal spectral
homogeneity among different types of cuprates [4,5,14-18].
In this work we investigate the effects of varying doping levels and
magnetic fields on the spatially resolved
low-energy quasiparticle excitations of
hole- and electron-type cuprate superconduc
tors. Our experimental results suggest
that the PG phenomena are closely related to COs, and that the correlation of
superconductivity (SC) and PG with different types of disorder may account for
the varying degrees of spatial homogeneity in the quasiparticle spectra. We also
demonstrate the effect of hole doping on th
e vortex-state irreversibility line, which
suggests the relevance of competing orders to vortex pinning.
2. Experimental
The primary experimental technique employed in this work is cryogenic scanning
tunneling spectroscopy (STS). Details
of the experimental setup, surface
preparations and methodology of data analysis for the STS studies have been
described elsewhere [5-7,14,15]. The hole-typ
e cuprates investigated in this work
include optimally and under-doped Y-123 single crystals with SC transition
temperatures
T
c
= 93 K, 85 K and 60 K, which correspond to hole doping levels of
p
= 0.15, 0.13 and 0.09, respectively; and over-doped (Y
1
x
Ca
x
)Ba
2
Cu
3
O
7

epitaxial films grown by pulsed laser deposition with
x
= 0.05, 0.10, 0.125, 0.20
and 0.30, and the corresponding
T
c
determined from magnetization measurements
were 68 K, 64 K, 59 K, 42 K and 74 K, respectively. We note that the
p
value for
a given Ca-doping level
x
depends on the oxygen annealing process [19], and that
there is a maximum
T
c
value for a given
x
,
T
c,max
(
x
), which were empirically
3
determined to be 93.5 K, 89.0 K, 82.9 K and 82.9 K for
x
= 0, 0.10, 0.20 and 0.30,
respectively [19]. Hence, the
p
values of Ca-doped Y-123 samples are estimated
from
T
c,max
(
x
) and the empirical formula [20]
T
c
(
x
,
p
) =
T
c,max
(
x
) [1
82.6(
p
0.16)
2
],
yielding
p
= 0.216, 0.218, 0.214, 0.238 and 0.19 for
x
= 0.05, 0.10, 0.125, 0.20
and 0.30. The electron-type cuprate studied in this work is an optimally doped
infinite-layer system Sr
0.9
La
0.1
CuO
2
(La-112) with
T
c
= 43 K [7,15]. All samples
had been characterized by x-ray diffract
ion and magnetization studies to ensure
single-phased structures and superconductivity. In addition to microscopic STS
studies, the effects of Ca-doping on macroscopic vortex dynamics were
investigated by measuring the irreversibility temperature
T
irr
(
H
,
x,p
) from the zero-
field-cool (ZFC) and field-cool (FC) magnetization (
M
) vs. temperature (
T
) curves
[21].
3. Doping Dependent Qu
asiparticle Tunneling
Spectra and Vortex Dynamics
Spatially resolved tunneling conductance (
dI
/
dV
)
vs. energy (
ω
=
eV
) spectra for
the quasiparticle local density of states (LDOS) maps at
T
= 6 K were obtained on
aforementioned Y-123, Ca-doped Y-123, and
La-112 samples in zero and finite
magnetic fields (
H
). For
H
= 0, the tunneling spectra revealed long-range spatial
homogeneity in under- and optimally doped Y-123 and optimally doped La-112
samples [4-7], which differ from the strong spatially inhomogeneous tunneling
spectra observed in Bi-2212 [17,18]. In contrast, for Ca-doped Y-123, the zero-
field LDOS spectra revealed spatial homogeneity only within a limited range (up
to ~ 10
2
nm in length); variations in the spectral characteristics appeared over a
long range, which may be attributed to disorder in Ca-doping.
3.1 Doping dependent zero-field LDOS of hole-type cuprates
A representative zero-field LDOS of the optimally doped Y-123 in the top panel
of Fig. 1a shows a set of coherent peaks at
ω
=

SC
and shoulder-like features at

eff
. Both features exhibit long-range spatial homogeneity, as manifested by the
histogram in the bottom panel of Fig. 1a. We attribute the two features to the
consequence of coexisting SC and CO in the ground state of the under- and
optimally doped hole-type cuprates [4-10]. Briefly, the LDOS
N
(
) is associated
4
with the spectral density function
A
(
k
,
) and the Green function
G
(
k
,
) by the
relation
N
(
) =
k
A
(
k
,
) =

k
Im[
G
(
k
,
)]/
, and
G
(
k
,
) may be obtained
from diagonalizing the mean-field Hamiltonian
H
MF
=
H
SC
+
H
CO
that consists
of coexisting SC and a CO, where
H
SC
is given by [4-10]

,SC
††
SC
,
,
,
,
,
,
cc
cc
c c

    


kk
k
k
k
k k
kk
k
H
.
Here the SC pairing potential is given by
SC
(
k
) =
d
(cos
k
x
cos
k
y
)/2 for pure
d
x
2
y
2
-wave pairing and
SC
(
k
) =
d
(cos
k
x
cos
k
y
)/2+
s
for (
d
x
2
y
2
+s)-wave pairing
[14],
k
denotes the quasiparticle momentum,
k
is the normal-state eigen-energy
relative to the Fermi energy,
c
and
c
are the creation and annihilation operators,
and
=
,
refer to the spin states. For
H
CO
, there is a CO energy
CO
and a
density wave-vector associated with a given CO [4-10]. In the case of charge
density waves (CDW) being the relevant CO, we have a
Q
1
parallel to the CuO
2
bonding direction (
,0)/(0,
) [4-10]. The LDOS thus obtained for the optimally
doped Y-123 is shown by the solid line in Fig. 1a, where
eff
[(
d
)
2
+(
CO
)
2
]
1/2
.
We further note the occasional occurre
nce of a zero-bias conductance peak
(ZBCP) for tunneling along the {100} direction, as exemplified in the main panel
of Fig. 1a, which is the result of the atomically rugged {100} surface so that
Andreev bound states near {110} can contribute to the tunneling spectra, as
detailed in Ref. [14].
For under-doped Y-123, the zero-field L
DOS also reveals similar spectral
features (Fig. 1b, upper panel), except that
d
is reduced and
eff
evolves from
shoulder-like features to peak-like features separated from the SC coherence
peaks. Both
d
and
eff
remain spatially homogeneous (Fig. 1b, lower panel).
In the case of Ca-doped Y-123, the pairing symmetry evolves from pure
d
x
2
y
2
to (
d
x
2
y
2
+
s
)-wave with
SC
(
k
) =
d
(cos
k
x
cos
k
y
)/2+
s
[14,22,23], and the
spectral characteristics are homogeneous only over smaller areas ~(10
2
10
2
) nm
2
probably due to disordered Ca-doping. As exemplified in Fig. 1c for a Ca-doped
Y-123 with
x
= 0.3 and
p
~ 0.19, two sets of coherent peaks appear at
=
(
d
+
s
) and
(
d

s
) [14], and the shoulder-like features correspond to
eff
=
[(
SC
)
2
+(
CO
)
2
]
1/2
, where
SC
max{
SC
(
k
)}. The doping dependent
SC
and
CO
5
for the Y-123 system is shown in Fig. 1d, showing a dome-like
SC
(
p
) similar to
that of
T
c
(
p
) and a decreasing
CO
(
p
) similar to the PG temperature
T
*
(
p
).
3.2 Vortex-state LDOS of hole-type cuprates
In the vortex state of conventional type
-II superconductors, SC inside the vortex
core is suppressed by the supercurrent
s surrounding each vortex, giving rise to
enhanced local density of states (LDOS) peaking at
= 0 near the center of each
vortex [24]. In contrast, the vortex-state LDOS of Y-123 exhibits several
important differences. First, despite spatially homogeneous zero-field LDOS,
field-induced vortices are relatively disordered and the radius of the vortex “halo”
(
halo
~ 10 nm) appears much larger than the SC coherence length
SC
~ 1.2 nm,
(Fig. 2a). Further, the vortex-state LDOS remains suppressed inside the vortex
core (Figs. 2c-2d), with PG-like feat
ures appearing at the same energy
CO
as that
derived from theoretical analysis of the
zero-field LDOS. Moreover, density-wave
like constant-bias conductance modulations are apparent, as exemplified in Fig.
2b. The histogram of the spectral evolution from
SC
to
CO
and another sub-gap
feature at

with increasing
H
is shown in Fig. 2e, which is in stark contrast to the
vortex-state spectral evolution of conv
entional type-II superconductors [4].
To obtain further insights, we perform Fourier transformation (FT) of the
LDOS at constant energies (
). As shown in Fig. 3a for the FT-LDOS in the
reciprocal space for spectra integrated from
1 to
30 meV, various spectral peaks
are apparent, which may be divided into two distinct types: One is associated with
the
-independent wave-vectors that may be attributed to COs of charge-, pair-
and spin-density waves (CDW, PDW and SDW) and the (
,
) magnetic
resonance, as shown in Figs. 3b, 3d and summarized in Fig. 3g [4-6]. The other
type consists of
-dependent quasiparticle inte
rference (QPI) wave-vectors
[5,6,17], as exemplified in Fig. 3e and summarized in Fig. 3f. The spectral
intensity of these
-independent wave-vectors exhibits interesting evolution with
H
that further corroborates the existence of COs, as exemplified in Fig. 3c [4-6].
3.3 LDOS of electron-type cuprates
The zero-field LDOS of electron-type c
uprate La-112 exhibited a single set of
spectral peaks at
ω
=

eff
(Fig. 4a), and the LDOS revealed long-range spatial
6
homogeneity [7]. Theoretical fitting to the LDOS and the k-dependent spectral
density from angle-resolved photoemission spectroscopy (ARPES) yields a
d
x
2
y
2
-
wave SC gap with
d
~ 12 meV and a SDW with a wave-vector of (
,
) and
SDW
~ 8 meV [4,5,7]. The vortex-state LDOS of La-112 revealed a vortex-core radius
comparable to
SC
~ 4.9 nm. The LDOS remained suppressed inside the vortex
core, with PG-like features appearing at
CO
<
SC
, as shown in Fig. 4c. The fact
that
CO
<
SC
is consistent with the absence of zero-field PG above
T
c
in La-112.
The histogram of the spectral evolution with
H
is illustrated in Fig. 4d, which
differs from those of Y-123 (Fig. 2e) an
d conventional type-II superconductors.
3.4 Doping dependent vortex dynamics
In addition to the LDOS, we investigate how vortex dynamics may evolve with
different doping. By applying
H
parallel to the CuO
2
planes of Y-123 and Ca-
doped Y-123 and determining the irreversibility temperatures
T
irr
(
H
,
p,x
) from the
ZFC and FC
M
-vs.-
T
curves [21], we find that the normalized irreversibility line
that separates the vortex-solid from the vortex-liquid initially decreases with
increasing Ca-doping
x
for nearly constant
p
(Fig. 5), suggesting suppressed SC
coherence due to Ca-induced disorder [23]. As the hole doping further increases,
the trend eventually reverses (Fig. 5), probably due to vanishing COs and
therefore enhanced SC stiffness and reduced vortex-state fluctuations [21].
4. Discussion
In addition to the doping dependence of
SC
and
eff
, it is interesting to address
the issue of spatial homogeneity of LDOS in different cuprates. Comparing our
empirical findings with the highly inhomo
geneous quasiparticle spectra in Bi-
2212 [17,18], it appears that the spatial homogeneity in the LDOS depends on the
type of disorder: For optimally and
under-doped Y-123, the long-range spatial
homogeneity in both
SC
and
eff
may be attributed to the stoichiometric cations
and ordered apical oxygen. In contrast,
for Ca-doped Y-123, the substitution of Y
by Ca contributes to excess holes as well as disorder in the CuO
2
planes, giving
rise to spatial variations in
SC
and
eff
. In the case of under and optimally doped
Bi-2212, the non-stoichiometric Bi-to-Sr ratio results in disordered Sr and apical
oxygen in the SrO layer [25], leading to highly disordered PG features. Finally,
7
for optimally doped electron-type cuprate La-112, the homogeneous LDOS may
be attributed to stoichiometric cations and the absence of apical oxygen.
5. Conclusion
Scanning tunneling spectroscopic studies
of various cuprate superconductors as a
function of doping level, doping type and magnetic field (
H
) reveal that their low-
energy excitations consist of not only
the Bogoliubov quasiparticles but also
bosonic excitations associated with
COs, leading to a PG above
T
c
for
H
= 0 and
inside the vortex core for
T
<<
T
c
if
CO
SC
, as in the under- and optimally
doped Y-123. In contrast, for
CO
<
SC
as in the electron-type cuprate La-112
and strongly over-doped Y-123, PG is absence above
T
c
for
H
= 0 and is only
revealed inside the vortex core for
T
<<
T
c
. The vortex core size
halo
for cuprates
with
CO
SC
is much larger than
SC
, whereas
halo
for cuprates with
CO
<
SC
is comparable to
SC
, suggesting that the vortex core states are sensitive to the CO
strength. The microscopic doping-dependent spectral characteristics are found to
be relevant to the macroscopic vortex dynamics, as manifested by the doping
dependent vortex irreversibility lines of Y-123.
We acknowledge funding provided by NSF Grant #DMR0907251, by the Institute for Quantum
Information and Matter, an NSF Physics Frontiers Center with support of the Gordon and Betty
Moore Foundation, and by the Kavli Nanoscience Institute with support of the Kavli Foundation.
ZJF acknowledges the support from China Scholarship Council during his visit to Caltech.
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Fig. 1:
(Color online) Doping dependent zero-field STS in Y-123 taken at
T
= 6 K:
(a)
Upper
panel: Normalized LDOS of optimally doped Y-123 (
T
c
= 93 K) with k || {100} and {001} (inset).
The solid lines represent theoretical fittings [4-10] by assuming coexisting SC and CDW with
parameters
SC
= 29 (21) meV for k || {100} ({001}),
CDW
= 32 meV and Q
CDW
= (0.25

0.05
,0)/(0,0.25

0.05
), and the zero-bias conductance peak in the main panel is due to
atomically rugged {100} surface so that some of the Andreev bound states near {110} are detected
and fit with the BTK theory [1
4]. Lower panel: Histograms for
SC
and
CDW
.
(b)
Upper panel:
Normalized LDOS of under-doped Y-123 (
T
c
= 60 K) with k || {100} and {001} (inset,
T
c
= 85 K).
Lower panel: Histograms for
SC
and
CDW
.
(c)
Normalized LDOS of Ca-doped Y-123 with k ||
{001},
T
c
= 74 K and
p
= 0.19. The parameters are (
d
,
s
,
CDW
) = (16, 3, 27) meV.
(d)
Zero-field
SC
and
CDW
of Y-123
vs.
p
.
9
Fig. 2:
(Color online) Spatially resolved STS studies of the vortex-state of Y-123 at
T
= 6 K [5,6]:
(a)
Tunneling conductance power ratio
r
G
map over a (75×40) nm
2
area for
H
= 4.5 T, showing
a
B
= (23.5
8.0) nm. Here
r
G
at each pixel is defined by the ratio of (
dI
/
dV
)
2
at
V
= (
SC
/
e
) to that at
V
= 0.
(b)
The LDOS modulations of Y-123 at
H
= 5 T over a (22×29)
nm
2
area, showing patterns
associated with density-wave modulations and vortices (circled objects) for
ω
=
9 meV
׽

and
ω
=
23 meV
׽

SC
.
(c)
Conductance spectra along the white line in (a), showing SC peaks
at
=

SC
outside vortices and PG features at
=

CO
inside vortices.
(d)
Spatially averaged
intra- and intervortex spectra for
H
= 2.0 T, 4.5 T and 6 T from left to right.
(e)
Energy histograms
for the field-dependent spectral weight derived from the STS data for
H
= 0, 2, 4.5, and 6T,
showing a spectral shift from
SC
to
CO
and

with increasing
H
.
(a)
(b)
(c)
(d)
(e)
CO
10
Fig. 3:
(Color online) Studies of the vortex-state FT-LDOS of Y-123 [4-6]:
(a)
FT-LDOS at
H
= 5
T obtained by integrating |F(
k
,
)| from
=

meV to
30 meV. The
-independent spots are
circled for clarity, which include the reciprocal lattice constants, the (
,
) resonance,
Q
PDW
and
Q
CDW
along the (
,0)/(0,
) directions, and
Q
SDW
along (
,
).
(b)
The
-dependence of |F
(
k
,
ω
)
| at
H
= 5 T is plotted in the
-
vs.
-
k
plot against
k
|| (
,0), showing
-independent modes (bright
vertical lines) at
Q
PDW
and
Q
CDW
.
(c)
|F(
q
,
)| for
q
=
Q
PDW
(red) and
Q
CDW
(green) are shown as a
function of
for
H
= 0 (solid lines) and
H
= 5 T (dashed lines).
(d)
|F(
k
,
)| for different energies
are plotted against
k
|| (
,0), showing peaks at
-independent Q
PDW
, Q
CDW
and the reciprocal
lattice constants at (2
/
a
1
) along (
,0).
(e)
|F(
k
,
)| for different energies are plotted against
k
||
(
,
), showing peaks at
-independent Q
SDW
along (
,
). Additionally, dispersive wave vectors
due to QPI are found, as exemplif
ied by the dispersive QPI momentum
q
7
specified in Fig. 3(f).
(f)
The QPI momentum (
|
q
i
|
) vs.
ω
dispersion relations derived fro
m FT-LDOS [4-6]. Lower panel:
Illustration of the
q
i
associated with QPI between pairs of points on equal energy contours with
maximum joint density of states.
(g)
Left panel: Illustration of the wave-vectors associated with
SDW and CDW. Right panel:
-independent
|
Q
PDW
|
,
|
Q
CDW
|
and
|
Q
SDW
|
[4-6].
(a)
(b) (c)
(d)
(e) (f)
(g)
SDW
11
Fig. 4:
(Color online) STS studies of La-112:
(a)
Normalized tunneling spectra taken at
T
= 6 K
(black) and 49 K (red) for
H
= 0. The solid lines represent fittings to the
T
= 6 and 49 K spectra by
assuming coexisting SC and SDW, with fitting parameters
d
= 12 meV,
CO
=
SDW
= 8 meV, and
Q
SDW
= (
,
) [4,5,7].
(b)
A spatial map of the
r
G
ratio over a (64×64) nm
2
area for
H
(||c) = 1 T and
T
= 6 K, showing vortices separated by an average vortex lattice constant
a
B
= 52nm, comparable
to the theoretical value of 49 nm. The average radius of vortices is
halo
= (4.7
0.7) nm,
comparable to
SC
= 4.9 nm.
(c)
Spatial evolution of (
dI
/
dV
) along the black dashed line cutting
through two vortices in (b) for
H
= 1T, showing an intra-vortex PG smaller than the inter-vortex
SC gap.
(d)
Energy histograms of La-112 (left), which differ from those of conventional type-II
SC (right).
(d)
(a) (b)
(c)
La-112
CO
12
Fig. 5:
Vortex irreversibility lines vs. (
T
/
T
c
) for
H
||
ab
in Y-123 systems, where the irreversibility
field
H
irr
(
T
,
x
,
p
) for each sample is normalized to its respective theoretical upper critical field
[
H
c2
(0)]
ab
by the relation [
H
c2
(0)]
ab
=
0
/[2
ab
(0)
c
(0)]. Using the mean-field gap relation 2
d
=
4.3
k
B
T
c
for
d
-wave pairing and the BCS relation
d
=
hv
F
/[2
ab
(0)], where
h
is the Plank constant
and
v
F
is the Fermi velocity (~10
5
m/s in the cuprates), we obtain [
H
c2
(0)]
ab
~ (0.18 Tesla/K
2
)
(
T
c
)
2
for
T
c
measured in K by using the empirical anisotropy ratio
=

ab
(0)/
c
(0) ~ 7 [21]. For
x
< 0.2,
the vortex solid phase below the normalized irreversibility line
H
irr
(
T
,
x
,
p
)/(0.18
T
c
2
) is suppressed
with increasing Ca-doping if
p
is kept nearly constant. However, for sufficiently large hole-doping
levels (such as for
p
0.23), the trend is reversed due to vanishing CO and increasing SC stiffness.