Macroscopic evidence for quantum criticality and field-induced quantum fluctuations in cuprate
superconductors
A. D. Beyer,
1
V. S. Zapf,
2
H. Yang,
1
F. Fabris,
2
M. S. Park,
3
K. H. Kim,
3
S.-I. Lee,
3
and N.-C. Yeh
1
1
Department of Physics, California Institute of Technology, Pasadena, California 91125, USA
2
National High Magnetic Field Laboratory, Los Alamos, New Mexico 87545, USA
3
Department of Physics, Pohang University of Science and Technology, Pohang, Korea
Received 15 January 2007; revised manuscript received 7 October 2007; published 25 October 2007
We present macroscopic experimental evidence for field-induced microscopic quantum fluctuations in dif-
ferent hole- and electron-type cuprate superconductors with varying doping levels and numbers of CuO
2
layers
per unit cell. The significant suppression of the zero-temperature in-plane magnetic irreversibility field relative
to the paramagnetic field in all cuprate superconductors suggests strong quantum fluctuations due to the
proximity of the cuprates to quantum criticality.
DOI:
10.1103/PhysRevB.76.140506
PACS number
s
: 74.25.Dw, 74.25.Op, 74.40.
k, 74.72.
h
High-temperature superconducting cuprates are extreme
type-II superconductors that exhibit strong thermal, disorder,
and quantum fluctuations in their vortex states.
1
–
9
While
much research has focused on the
macroscopic
vortex dy-
namics of cuprate superconductors with phenomenological
descriptions,
1
–
5
,
7
little effort has been made to address the
microscopic
physical origin of their extreme type-II nature.
9
Given that competing orders
COs
can exist in the ground
state of these doped Mott insulators besides superconductiv-
ity
SC
,
9
–
15
the occurrence of quantum criticality may be
expected.
11
,
13
,
16
The proximity to quantum criticality and the
existence of COs can significantly affect the low-energy ex-
citations of the cuprates due to strong quantum fluctuations
8
,
9
and the redistribution of quasiparticle spectral weight among
SC and COs.
9
,
17
,
18
Indeed, empirically the low-energy exci-
tations of cuprate superconductors appear to be unconven-
tional, exhibiting intriguing properties unaccounted for by
conventional Bogoliubov quasiparticles.
9
,
17
–
19
Moreover, ex-
ternal variables such as temperature
T
and applied mag-
netic field
H
can vary the interplay of SC and COs, such as
inducing or enhancing
20
,
21
the COs at the price of more rapid
suppression of SC, thereby leading to weakened supercon-
ducting stiffness and strong thermal and field-induced
fluctuations.
1
–
3
On the other hand, the quasi-two-dimensional
nature of the cuprates can also lead to quantum criticality in
the limit of decoupling of CuO
2
planes.
6
In this work we
demonstrate experimental evidence from macroscopic mag-
netization measurements for field-induced quantum fluctua-
tions among a wide variety of cuprate superconductors with
different microscopic variables such as the doping level
of holes or electrons, and the number of CuO
2
layers per unit
cell
n
.
22
We suggest that the manifestation of strong field-
induced quantum fluctuations is consistent with a scenario
that all cuprates are in close proximity to a quantum critical
point
QCP
.
6
To investigate the effect of quantum fluctuations on the
vortex dynamics of cuprate superconductors, our strategy in-
volves studying the vortex phase diagram at
T
→
0 to mini-
mize the effect of thermal fluctuations, and applying mag-
netic field parallel to the CuO
2
planes
H
ab
to minimize
the effect of random point disorder. The rationale for having
H
ab
is that the intrinsic pinning effect of layered CuO
2
planes generally dominates over the pinning effects of ran-
dom point disorder, so that the commonly observed glassy
vortex phases associated with point disorder for
H
c
e.g.,
vortex glass and Bragg glass
1
,
5
,
7
can be prevented. In the
absence of quantum fluctuations, random point disorder can
cooperate with the intrinsic pinning effect to stabilize the
low-temperature vortex smectic and vortex solid phases,
4
so
that the vortex phase diagram for
H
ab
would resemble that
of the vortex-glass and vortex-liquid phases observed for
H
c
with a glass transition
H
G
T
=0
approaching
H
c
2
T
=0
. On the other hand, when field-induced quantum fluc-
tuations are dominant, the vortex phase diagram for
H
ab
will deviate substantially from predictions solely based on
thermal fluctuations and intrinsic pinning, and we expect
strong suppression of the magnetic irreversibility field
H
irr
ab
relative to the upper critical field
H
c
2
ab
at
T
→
0, because the
induced persistent current circulating along both the
c
axis
and the
ab
plane can no longer be sustained if field-induced
quantum fluctuations become too strong to maintain the
c
-axis superconducting phase coherence.
In this Rapid Communication we present experimental re-
sults that are consistent with the notion that all cuprate su-
perconductors exhibit significant field-induced quantum fluc-
tuations as manifested by a characteristic field
H
irr
ab
T
→
0
H
*
H
c
2
ab
T
→
0
. Moreover, we find that we can express
the degree of quantum fluctuations for each cuprate in terms
of a reduced field
h
*
H
*
/
H
c
2
ab
0
, with
h
*
→
0 indicating
strong quantum fluctuations and
h
*
→
1 referring to the
mean-field limit. Most important, the
h
*
values of all cu-
prates appear to follow a trend on an
h
*
vs.
plot, where
is a material parameter for a given cuprate that reflects its
doping level, electronic anisotropy, and charge imbalance if
the number of CuO
2
layers per unit cell
n
satisfies
n
3.
23
,
24
In the event that
H
c
2
ab
0
exceeds the paramagnetic field
H
p
SC
0
/
2
B
for highly anisotropic cuprates, where
SC
0
denotes the superconducting gap at
T
=0,
h
*
is de-
fined by
H
*
/
H
p
because
H
p
becomes the maximum critical
field for superconductivity.
To find
h
*
, we need to determine both the upper critical
field
H
c
2
ab
T
and the irreversibility field
H
irr
ab
T
to as low a
temperature as possible. Empirically,
H
c
2
ab
T
can be derived
from measuring the magnetic penetration depth in pulsed
fields, with
H
c
2
ab
0
extrapolated from
H
c
2
ab
T
values obtained
at finite temperatures. The experiments involve measuring
PHYSICAL REVIEW B
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RAPID COMMUNICATIONS
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/140506
4
©2007 The American Physical Society
140506-1
the frequency shift
f
of a tunnel diode oscillator
TDO
resonant tank circuit with the sample contained in one of the
component inductors. Details of the measurement techniques
have been given in Ref.
8
. In general, we find that the con-
dition
H
c
2
ab
0
H
p
is satisfied among all samples investi-
gated so that we define
h
*
H
*
/
H
p
hereafter. On the other
hand, determination of
H
c
2
ab
0
and
H
c
2
c
0
is still useful be-
cause it provides the electronic anisotropy
ab
/
c
=
H
c
2
ab
0
/
H
c
2
c
0
, where
ab
c
refers to the in-plane
c
-axis
superconducting coherence length.
To determine
H
irr
ab
T
, we employed different experimental
techniques, including dc measurements of the magnetization
M
T
,
H
with the use of a superconducting quantum interfer-
ence device SQUID magnetometer or a homemade Hall
probe magnetometer for lower fields
up to 9 T
, a dc mag-
netometer
up to 14 T
at the National High Magnetic Field
Laboratory
NHMFL
in Los Alamos
LANL-PPMS
, a can-
tilever magnetometer at the NHMFL in Tallahassee for
higher fields
up to 33 T dc fields in a
3
He refrigerator
,
25
and a compensated coil for magnetization measurements in
the pulsed-field facilities at LANL for even higher fields
up
to 65 T pulsed fields in a
3
He refrigerator
.
8
In addition, ac
measurements of the third-harmonic magnetic susceptibility
3
as a function of temperature in a constant field are em-
ployed to determine the onset of nonlinearity in the low-
excitation limit.
26
Examples of the measurements of
H
irr
ab
T
for
HgBa
2
Ca
2
Cu
3
O
x
Hg-1223,
T
c
=133 K
,
HgBa
2
Ca
3
Cu
4
O
x
Hg-1234,
T
c
=125 K
, HgBa
2
Ca
4
Cu
5
O
x
Hg-1245,
T
c
=108 K
, and Sr
0.9
La
0.1
CuO
2
La-112,
T
c
=43 K
, are shown in Figs.
1
a
–
1
d
, and the consistency
among
H
irr
ab
T
deduced from different techniques have been
verified, as summarized in the
H
vs
T
phase diagrams
H
ab
in Figs.
2
a
–
2
d
. The Hg-based cuprates are in ei-
ther polycrystalline or grain-aligned forms, and the quality of
these samples is confirmed with x-ray diffraction and mag-
netization measurements to ensure single phase and nearly
100% volume superconductivity.
27
,
28
We have also verified
that
H
irr
ab
T
obtained from the polycrystalline samples are
consistent with those derived from the grain-aligned samples
with
H
ab
, because the measured irreversibility in a poly-
crystalline sample is manifested by its maximum irreversibil-
FIG. 1. Representative measure-
ments of the in-plane irreversibility
fields
H
irr
ab
T
in cuprate superconduct-
ors:
a
Hg-1223
polycrystalline and
grain aligned
,
b
Hg-1234
polycrys-
talline
,
c
Hg-1245
grain aligned
,
and
d
La-112
polycrystalline and
grain aligned
. Insets:
a
Consistent
T
irr
ab
H
obtained from maximum irre-
versibility of a polycrystalline sample
and from irreversibility of a grain-
aligned sample with
H
ab
;
b
repre-
sentative
3
data taken using ac Hall
probe techniques
Ref.
26
;
c
details
of the
M
vs
T
curves, showing an
anomalous upturn at
T
T
̃
;
d
exem-
plifying determination of
H
c
2
ab
in La-112
using TDO to measure
f
Ref.
8
.
FIG. 2. Determination of
H
*
using
various magnetic measurements of
H
irr
ab
T
:
a
Hg-1223,
b
Hg-1234,
c
Hg-1245, and
d
La-112. In
b
and
d
dashed lines indicate
H
c
2
ab
T
from TDO
measurements. In
d
we also illustrate
H
c
2
c
for comparison. We note reason-
able consistency among different ex-
perimental techniques, indicating
strong suppression of
H
*
H
irr
ab
0
rela-
tive to
H
c
2
ab
0
or
H
p
in all cuprates.
BEYER
et al.
PHYSICAL REVIEW B
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RAPID COMMUNICATIONS
140506-2
ity
H
irr
ab
among grains of varying orientation relative to the
applied field. Examples of this consistency have been shown
in Ref.
8
and also in the main panel and the inset of Fig.
1
a
.
In addition to the four different cuprates considered in this
work, we compare measurements of
H
irr
ab
T
on other cuprate
superconducting single crystals, including underdoped
YBa
2
Cu
3
O
7−
Y-123,
T
c
=87 K
,
29
optimally doped
Nd
1.85
Ce
0.15
CuO
4
NCCO,
T
c
=23 K
,
30
and overdoped
Bi
2
Sr
2
CaCu
2
O
x
Bi-2212,
T
c
=60 K
.
31
The irreversibility
fields for these cuprates normalized to their corresponding
paramagnetic fields
H
p
are summarized in Fig.
3
a
as a
function of the reduced temperature
T
/
T
c
, clearly demon-
strating strong suppression of
H
*
relative to
H
p
and
H
c
2
ab
0
in
all cuprates and implying significant field-induced quantum
fluctuations.
The physical significance of
h
*
may be better understood
by considering how the magnetic irreversibility for
H
ab
occurs. For sufficiently low
T
and small
H
, a supercurrent
circulating both along and perpendicular to the CuO
2
planes
with a coherent superconducting phase can be induced and
sustained, leading to magnetic irreversibility. On the other
hand, strong thermal or quantum fluctuations due to large
anisotropy and competing orders in the cuprates can reduce
the phase coherence of supercurrents, particularly the coher-
ence perpendicular to the CuO
2
planes, thereby diminishing
the magnetic irreversibility. Thus, we expect that the degree
of in-plane magnetic irreversibility is dependent on the
nominal doping level
, the electronic anisotropy
, the num-
ber of CuO
2
layers per unit cell,
n
, and the ratio of charge
imbalance
23
,
24
o
/
i
between the doping level of the outer
layers
o
and that of the inner layer
s
i
in multilayer
cuprates with
n
3. In other words,
h
*
for each cuprate su-
perconductor may be expressed in terms of a material param-
eter
that depends on
,
,
n
, and
o
/
i
, and the simplest
assumption for a linearized dependence of
on these vari-
ables gives
−1
o
/
i
−
n
−2
n
3
,
1
−1
n
2
.
2
If the suppression of the in-plane magnetic irreversibility is
associated with field-induced quantum fluctuations and the
proximity to a quantum critical point
c
,
11
h
*
should be a
function of
−
c
. Indeed, we find that using the empiri-
cally determined values for different cuprates tabulated in
Table
I
and the definition of
given above, the
h
*
vs.
data
for a wide variety of cuprates appear to follow a trend, as
shown in Fig.
3
b
. For comparison, we include in Fig.
3
b
theoretical curves predicted for field-induced static spin den-
sity waves
SDWs
in cuprate superconductors in the limit of
H
c
1
0
H
H
c
2
0
, where
h
*
above which static SDWs
coexist with SC
satisfies the relation
11
FIG. 3.
a
Reduced in-plane fields
H
irr
ab
/
H
p
and
H
c
2
ab
/
H
p
vs
T
/
T
c
for various cuprates. In
the
T
→
0 limit where
H
irr
ab
→
H
*
, the reduced
fields
h
*
H
*
/
H
p
1 for all cuprates are listed
in Table
I
for Y-123, NCCO, Bi-2212, La-112,
Hg-1234, Hg-1223, and Hg-1245
in descending
order
.
b
h
*
vs
in logarithmic plot for differ-
ent cuprates, with decreasing
representing in-
creasing quantum fluctuations. The lines given by
−400
−
c
/ln
−
c
represent the field-induced
SDW scenario
Ref.
11
in Eq.
3
with different
c
=0, 10
−4
, and 2
10
−4
from left to right. Inset:
Linear plot of the main panel.
c
The same data
as in
b
are compared with the power-law depen-
dence
solid lines
given by 5
−
c
1/2
, using
different
c
=0, 10
−4
, and 2
10
−4
from left to
right. Inset: Linear plot of the main panel.
d
H
vs.
T
diagram of Hg-1245.
See text for details.
TABLE I. Quantum criticality parameters among different cuprates. All fields in tesla.
denotes a parameter’s uncertainty.
Compound
o
i
10
−2
10
−3
H
*
H
*
H
c
2
ab
H
P
H
P
h
*
h
*
Hg-1245
0.15 1.30 0.80 55
Ref.
32
25
0.06
0.3
23.0 5.0
278
40 0.08 0.02
Hg-1223
0.15 1.04 0.92 52
Ref.
33
18
0.26
0.9
48.5 6.5
347
50 0.14 0.02
Hg-1234
0.15 1.20 0.80 52
Ref.
33
10
0.13
0.2
75.0 10.0
320
46 0.23 0.02
La-112
0.10 1.00 1.00 13
Ref.
8
4.0
0.77
2.4
46.0 4.0 160
110
10 0.42 0.04
Bi-2212
0.225 1.00 1.00 11
Ref.
31
8.0
2.05
15
65.0 10 100
155
22 0.42 0.06
NCCO
0.15 1.00 1.00 13
Ref.
30
5.0
1.15
4.4
40.0 5.0
77
59
8.0 0.68 0.12
Y-123
0.13 1.00 1.00 7.0
Ref.
33
2.0
1.86
5.3
210
50 600
239
25 0.88 0.23
MACROSCOPIC EVIDENCE FOR QUANTUM CRITICALITY
...
PHYSICAL REVIEW B
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RAPID COMMUNICATIONS
140506-3
h
*
−
c
/
ln
−
c
.
3
Here
c
is a nonuniversal critical point,
11
h
*
→
0 for
→
c
, and we have shown theoretical curves associated with
three different
c
values for comparison with data. On the
other hand, a simple scaling argument would assert a power-
law dependence:
h
*
−
c
a
a
0
.
4
Using
a
0.5 in Eq.
4
, we compare the power-law depen-
dence with experimental data in Fig.
3
c
. This dependence
appears to agree better with experimental data than the
SDW-SC formalism in Eq.
3
.
Although our available data cannot accurately determine
c
, we further examine Hg-1245
which has the smallest
h
*
for additional clues associated with the nature of the QCP.
We find that the magnetization
M
of Hg-1245 always exhib-
its an anomalous increase for
T
T
̃
H
see the inset of Fig.
1
c
, indicating a field-induced reentry of magnetic ordering
below
T
̃
H
. This magnetism reentry line
H
̃
T
is shown to-
gether with
H
irr
ab
T
in Fig.
3
d
. We suggest that the regime
below both
H
irr
ab
and
H
̃
corresponds to a coherent SC state
c
-SC
, and that bounded by
H
irr
ab
and
H
̃
is associated with a
coherent phase of coexisting SC and magnetic CO
c
-SC–CO
, whereas that above
H
irr
ab
is an incoherent SC
phase
i
-SC and
i
-SC–CO
with strong fluctuations.
Our conjecture of a field-induced magnetic CO in Hg-
1245 contributing to quantum fluctuations may be further
corroborated by considering the
h
*
vs.
dependence in the
multilayered cuprates Hg-1223, Hg-1234, and Hg-1245.
While these cuprate superconductors have the highest
T
c
and
H
c
2
values, as shown in Table
I
, they also exhibit the smallest
h
*
and
values, suggesting maximum quantum fluctuations.
These strong quantum fluctuations can be attributed to both
their extreme two-dimensionality
27
,
34
i.e., large
and sig-
nificant charge imbalance that leads to strong CO in the inner
layers.
23
,
24
Indeed, muon spin resonance
SR
experiments
35
have revealed increasing antiferromagnetic or-
dering in the inner layers of the multilayer cuprates with
n
3. Given that the
values of all Hg-based multilayer cu-
prates are comparable
Table
I
, the finding of larger quan-
tum fluctuations
i.e., smaller
h
*
in Hg-1245 is suggestive of
increasing quantum fluctuations with stronger competing or-
der. However, further investigation of the
h
*
and
values of
other multilayer cuprates will be necessary to confirm
whether competing orders in addition to large anisotropy
contribute to quantum fluctuations.
In summary, our investigation of the in-plane magnetic
irreversibility in a wide variety of cuprate superconductors
reveals strong field-induced quantum fluctuations. The mac-
roscopic irreversibility field exhibits dependences on such
microscopic material parameters as the doping level, the
charge imbalance in multilayered cuprates, and the electronic
anisotropy. Our finding is consistent with the notion that cu-
prate superconductors are in close proximity to quantum
criticality.
Research at Caltech was supported by NSF Grant No.
DMR-0405088 and through the NHMFL. The SQUID data
were taken at the Beckman Institute at Caltech. Work at Po-
hang University was supported by the Ministry of Science
and Technology of Korea. The authors gratefully acknowl-
edge Kazuyasu Tokiwa and Tsuneo Watanabe at the Tokyo
University of Science for providing the HgBa
2
Ca
4
Cu
5
O
x
Hg-1245
samples.
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