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Cooper Pair Breaking 251
where ω
ex
= 5 meV and β = 0.8.
74
The size of the coupling constant I follows
from the heavy-quasiparticle masses in that compound. The equations have
a solution for an order parameter of the form (62). The sin(cp
z
) solution can
be excluded, because it requires a spin triplet and gives a lower T
c
than the
cos(cp
z
) solution. For the latter a value of T
c
' 3 K is found which is of the
right order.
51
Note that there is no solution existing for an order parameter
of s-wave symmetry.
Why is s-wave pairing excluded while an order parameter of the form of
(62) gives a seizable T
c
? Pair formation is not expected since excitations
of magnetic excitons violate time-reversal symmetry. Therefore they break
Cooper pairs of electrons in time reversed states. However, when the order
parameter is of the form of (62) electrons are paired in states which are
time reversed followed by a lattice vector translation. Expressed differently,
electrons forming a pair are located preferably on different sublattices.
Therefore magnetic excitons can provide a pair forming electron interaction
here.
The two examples of PrOs
4
Sb
12
and UAl
2
Pd
3
round off our understand-
ing of pair breaking and its relation to time-reversal symmetry.
Acknowledgments
Part of this chapter is based on Ref. 27. I would like to thank G. Zwicknagl
and P. Thalmeier for numerous stimulating discussions and T. Takimoto for
careful reading of the manuscript.
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August 24, 2010 17:25 World Scientific Review Volume - 9.75in x 6.5in ch12
SUPERCONDUCTOR-INSULATOR TRANSITIONS
A. M. Goldman
School of Physics and Astronomy, University of Minnesota,
116 Church St. SE, Minneapolis, MN 55455, USA
goldman@physics.umn.edu
Superconductor-insulator transitions, especially in thin films, can provide
the simplest examples of the continuous quantum phase transition
paradigm. Quantum phase transitions differ from thermal phase transi-
tions in that they occur at zero temperature when the ground state of
a system is changed in response to a variation of an external parame-
ter of the Hamiltonian. In the example of the superconductor-insulator
transition, this control parameter could be the parallel or perpendicular
magnetic field, disorder, or charge density. Quantum phase transitions
are studied through measurements at nonzero temperature of physical be-
haviors influenced by the quantum fluctuations associated with the tran-
sition. This review will focus on experimental aspects of superconductor-
insulator transition in disordered films that are effectively two-dimensional.
In particular, the evidence for quantum critical behavior in the various
types of transitions will be presented. The various theoretical scenarios for
the transitions will also be discussed along with the extent to which they
are supported by experiment. Open questions relating to the nature of
the very puzzling insulating regime and whether there are many different
types of superconductor-insulator transitions will be presented. Although
this research area is more than 20 years old, rather central issues are not
resolved.
1. Introduction
In this short review we will focus on experimental aspects of superconductor-
insulator (SI) transitions that have been studied in disordered ultrathin films
that are effectively two-dimensional (2D) or quasi-2D. These provide the sim-
plest examples of SI transitions, which are part of an important paradigm
of contemporary condensed matter physics, the continuous quantum phase
255
August 24, 2010 17:25 World Scientific Review Volume - 9.75in x 6.5in ch12
256 A. M. Goldman
transition (QPT).
1
We will not discuss three-dimensional systems
2
or junc-
tion networks,
3
which have also been studied, and which in the case of
junction networks can serve as model systems for films.
Continuous QPTs occur at zero temperature, when the ground state of
a system is changed in response to the variation of an external parameter
of the Hamiltonian. In the example of the SI transition this could be par-
allel or perpendicular magnetic field, disorder, or charge density. Quantum
phase transitions are studied through measurements at nonzero tempera-
ture of physical behaviors influenced by quantum fluctuations. Among other
systems exhibiting quantum phase transitions are
4
He adsorbed on random
substrates, two-dimensional electron gases, and numerous complex strongly
correlated electron materials including high temperature superconductors.
The concept is also relevant to the physics of cold atoms, which in some in-
stances can serve as model systems for many condensed matter phenomena.
4
Superconductor-insulator transitions are of particular importance be-
cause of their connection with the fundamental phase-number uncertainty
relation, which makes them the simplest of such transitions. Although the
subject of SI transitions might be considered to be mature, there remain
many open issues resulting from the study of new materials, the application
of new experimental techniques, and the extension of measurements to lower
temperatures and higher magnetic fields.
2. Materials and Tuning Parameters
Experiments on various materials with different tuning parameters do not
clearly distinguish between competing theoretical pictures. Indeed there may
be several types of SI transitions. First, the mechanism may depend upon
the tuning parameter, which can be disorder, electrostatic charge, perpen-
dicular or parallel magnetic field, surface magnetic impurities, or dissipation.
Second, the mechanism and phase diagram may depend upon the type of film
being investigated, i.e. whether it is an ultrathin quench condensed amor-
phous or granular film, an amorphous film of InO
x
, MoGe, Nb
0.15
Si
0.85
, or
Ta, or a polycrystalline film of TiN. Third, the character of the substrate
may matter. The nature of the transition could be altered in the presence
of a high dielectric constant substrate such as SrTiO
3
, or may depend upon
the nature of an underlayer. Finally another issue is the role of spin-orbit
scattering. Experiments on high-Z elements (a-Bi or a-Pb) may produce
different results from those carried out on low-Z elements (a-Be or a-Al),
where there is minimal spin-orbit scattering.
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Superconductor-Insulator Transitions 257
Among some of the phenomena that have sparked renewed interest in
the field are the giant magnetoresistance peak found in some nearly critical
SI transition systems, leading in some instances to what has been called
“superinsulating” behavior at sufficiently low temperatures.
5,6
In some sys-
tems there appears to be a “quantum metal” in the limit of very high mag-
netic fields.
7,8
A deep analogy may exist between the insulating regime of
some of these systems and the pseudogap regime of the high temperature
superconductors.
9,10
Other issues include the possibility of an intermediate
metallic phase between the superconductor and insulator in some perpen-
dicular magnetic field tuned transitions.
11
The nature of the insulating regime, apart from the magnetoresistance
peak, is an important issue, as there is new experimental evidence from a
variety of sources that superconducting coherence persists over finite length
scales.
12
–
15
Also, the Nernst effect has emerged as a means of exploring
the insulating regime of the field-tuned SI transition.
16
–
18
The possibility of
electrostatically inducing superconductivity has been demonstrated, allow-
ing the exploration of behavior that does not depend on possible changes
in morphology associated with different doping or film thickness.
19,20
Using
electrostatic charging or doping it is possible to import into the study of the
SI transition some of the approaches used for 2D electron gas physics.
3. Theoretical Scenarios
Interest in SI transitions arose in the context of study of the interplay
between superconductivity and disorder. Anderson
21
and Abrikosov and
Gor’kov
22
showed that nonmagnetic impurities do not affect the supercon-
ducting transition because Cooper pairs form from extended time-reversed
eigenstates, which have disorder included. On the other hand with sufficient
disorder Anderson localization occurs, and superconductivity will not occur,
even in the presence of an attractive electron–electron attraction.
23
This
leads to the question of how superconductivity disappears with increasing
disorder.
24
At nonzero temperatures, the superconducting phase transition
is of second order, so that there are fluctuations. For a transition suppressed
to zero temperature, the fluctuations become quantum in character and the
transition becomes a quantum phase transition.
1
In two dimensions (2D) there is added complication as the supercon-
ducting transition is a topological, or Kosterlitz–Thouless–Berezinskii tran-
sition.
25
Initially, all 2D electronic systems were believed to be localized,
even for arbitrarily weak disorder. This changed with the observation of
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258 A. M. Goldman
apparent metal-insulator transitions in strongly interacting high mobility
low-carrier-density MOSFETs.
26
Superconductor-insulator transitions in 2D
are additional examples in which interactions play a critical role.
There have been four general approaches to explaining the demise of su-
perconductivity in the 2D limit as a function of disorder or magnetic field.
The first is based on a perturbative microscopic description of homogeneous
systems and considers the interplay between the attractive and repulsive
electron–electron interactions in the presence of disorder. In this microscopic
description, suppression of the critical temperature follows from the renor-
malization of the electron–electron interaction in the Cooper channel by the
long range Coulomb repulsion in the presence of disorder.
27
–
29
The ampli-
tude of the superconducting order parameter is suppressed by disorder, and
at a sufficient level of disorder, Cooper pairs break up into fermions, which
localize in 2D. The second is a boson localization theory, which predicts a
quantum critical point. Inside the critical region, Cooper pair phase fluctu-
ations are important, and these pairs in some approximation are bosons.
30
The observation of apparently direct SI transitions in ultrathin films, in the
T → 0 limit
31,32
in part served to motivate this approach. This will be
discussed below. The third is based on the physics of granular superconduc-
tors. It builds on dissipative models of resistively shunted Josephson junction
arrays, and may have application to both granular and homogeneous 2D
superconductors. The fourth involves percolation. The message conveyed
by experiment regarding the applicability of various models continues to be
quite mixed and indeed the explanation of some transitions may involve more
than one idea.
3.1. The Fermion localization scenario
As long as the reduction of the transition temperature is relatively small, the
perturbative approach mentioned above should provide an accurate expres-
sion for the data. In the regime where the transition would be suppressed
to zero there will be quantum fluctuations, and this theory should fail, as
such fluctuations are not included.
33
On the other hand, the reduction of
the mean field transition temperature that is predicted by perturbative the-
ory based on the weakening of the screening of the Coulomb interaction with
increasing disorder
29
is consistent with studies of homogeneous a-MoGe films
by Graybeal and Beasley
34
(see Fig. 1).
3.2. The Boson localization scenario
Boson localization theories
30,35,36
are closely related to models used to
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Superconductor-Insulator Transitions 259
Fig. 1. Suppression of superconductivity in amorphous Mo
x
Ge
y
with increasing
disorder.
34
The solid line is a theoretical fit as discussed in Ref. 29.
describe superfluid helium and Josephson junction arrays. Doniach
37
pio-
neered this approach involving phase fluctuations, in the context of Joseph-
son junction arrays. There is a quantum critical point, and the ground state
can be controlled by disorder or perpendicular magnetic field. An important
prediction is that at a critical level of disorder, or magnetic field, Cooper
pairs move diffusively, leading to metallic behavior in the T → 0 limit only
at the quantum critical point. If one assumes complete duality between
Cooper pairs and vortices
38
this resistance is universal with a value given by
the quantum resistance for pairs, h/4e
2
. The physics is related to the univer-
sal jump condition in the finite temperature Kosterlitz–Thouless–Berezinskii
transition.
25
The first real indication that disorder might drive the transition tempera-
ture down to a low enough value that quantum fluctuations could be impor-
tant was the work of Haviland, Liu and Goldman
31
in which the thickness
variation of resistance versus temperature R(T ), was studied in amorphous
Bismuth (a-Bi) films. Bismuth is a semimetal in its usual form, but is metal-
lic and superconducting under high pressure, or in thin film amorphous form,
which results when it is grown on substrates held at liquid helium temper-
atures.
39
This investigation was carried out in an apparatus in which the
films could be grown in situ, at low temperatures and measurements carried
out in alternation with increments of growth. If such studies are carried out
using substrates pre-coated with a thin layer of amorphous Ge or Sb, atomic
scale rather than mesoscale disorder apparently results.
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260 A. M. Goldman
In contrast, films made using a low temperature evaporation technique
but without an underlayer of amorphous Ge or Sb, are granular in that they
possess mesoscale clusters. They also exhibit a separation between super-
conducting and insulating behavior in the limit of zero temperature, but
their properties differ in detail from the nominally homogeneous amorphous
films described above. In particular, the dependence of the resistance versus
temperature is a nonmonotonic function of temperature.
40
Curves of R(T ) for a sequentially deposited set of a-Bi films grown on
a-Ge substrates are presented in Fig. 2. These resemble renormalization
flows to an unstable fixed point at zero temperature, which for this partic-
ular set of films correspond to the quantum resistance for pairs, h/4e
2
, or
6450 Ω/. This result set the stage for consideration of the SI transition
as a continuous quantum phase transition and motivated the application of
Boson physics to the SI transition. Unfortunately subsequent studies of the
SI transition in other materials indicated that the critical resistance was not
universally h/4e
2
. Yazdani and Kapitulnik
41
in their study of the field-driven
SI transition, suggested that the nonuniversal critical point resistance could
be a consequence of a parallel fermionic conductance channel at nonzero
temperature, a view also supported by Gantmakher and collaborators.
42
However this approach does not describe all of the systematics of the data
of Markovic et al.,
43,44
on the field-driven transition of quench-evaporated
films of different thicknesses.
Without duality assumption, and with a realistic form for the interac-
tion between charges, the resistance at the critical point is a nonuniversal
constant. Experimentally the resistance appears to be sample-dependent
but is within a factor of 2 or 1/2 of the quantum resistance for pairs,
h/4e
2
= 6450 Ω,
45
although recent work
46,47
suggests that there may be
regimes in which the critical resistance of the field-driven SI transition may
indeed be h/4e
2
. In elaborations of the Bose-Hubbard model, the value of
the critical resistance depends on the form of interaction between charges,
and the details of the calculations. (See for example Refs. 48–53.) Some of
these works are concerned with including low energy quasiparticle interac-
tions in the theory. These are claimed to be a most serious aspect of missing
physics, and they introduce amplitude fluctuations into the description of
the transition.
Tunneling experiments on quench evaporated homogeneous films of met-
als similar in preparation to those shown in Fig. 2 seem to support fermion
rather than boson scenarios.
54
The mean field transition temperature and
the energy gap are suppressed at a fixed ratio as sheet resistance increases,