FEATURE ARTICLE
THE
BY
SUPERCONDUCTING ENERGY GAP
TOM TIMUSK
T
he superconducting gap can be defined as the
energy difference between the ground state of the
superconductor and the energy of the lowest
quasiparticle excitation [1]. There were early hints
that such a gap existed but the first experimental evidence
for a gap came from the temperature dependence of the
specific heat below the transition temperature Tc as
measured by Corak et al. [2]. It was found that the
electronic specific heat was given by Ce%γTce-1.5Tc/T
where γ is the normal state electronic specific heat
coefficient and Tc the superconducting transition
temperature. The first spectroscopic measurement of the
energy gap was carried out with microwaves by Biondi
et al [3] on aluminum and far infrared techniques by
Glover et al [4] on lead. At the same time, the microscopic
theory of superconductivity, the BCS theory, was
announced by Bardeen, Cooper and Schrieffer [5] who
predicted the value of the ratio of the superconducting gap
to the transition temperature to be 1.76 which is in
excellent agreement with the spectroscopic measurements.
This was one of the earliest triumphs of the theory. It
should be noted that this value of the gap is from the weak
coupling limit of the theory and applies to materials with
For conventional
low Tc's such as aluminum.
superconductors, tunnelling spectroscopy [6] has been a
popular tool for gap measurement but for the new high
temperature superconductors, with their larger gaps,
infrared and photoemission spectroscopies have played an
increasingly important role.
A superconducting gap has a profound influence on the
response of a superconductor to an alternating
electromagnetic field. An incoming photon can be
absorbed only if its energy and momentum can be
transferred to the superconductor. Energy conservation
SUMMARY
A gap at the Fermi level is a signature
property of the conventional BCS
superconductors. High temperature superconductors have a gap with nodes but in
addition, at low doping levels, there is a
normal state gap called a pseudogap.
Whether this gap is a result of a precursor
state to superconductivity or a completely
separate order that competes with
superconductivity is an open question.
Fig. 1
Inverse square of the superconducting penetration depth
as function of temperature from Hardy et al [11]. The
linear T dependence at low temperature is evidence for
a d-wave order parameter in contrast to the s-wave
exponential dependence shown as the solid line.
demands that the photon energy hν > 2Δ, twice the energy
gap. The factor two comes from the destruction of a
Cooper pair to create a pair of quasiparticles. A second
requirement is the conservation of momentum. By
creating a bosonic excitation, typically a phonon or a spin
fluctuation, or by elastic scattering with a static defect, one
of the quasiparticles can be scattered away from the Fermi
surface and its momentum transferred from the
superconducting condensate to other degrees of freedom.
For elastic scattering to dominate, the scattering rate has to
be high enough to reach the dirty limit, defined as 1/τ > 2Δ
where τ is the quasiparticle life time. In the dirty limit
there is a sharp onset of absorption for frequencies greater
than 2Δ and the optical method is a good way of
determining the energy gap. On the other hand, if the
clean limit applies, in particular in the extreme case where
1/τ 2Δ, additional energy has to be supplied to create
the bosonic excitation and the onset of absorption occurs
at hν > 2Δ + ΩE for a single Einstein boson at ΩE.
According to a theorem of Anderson [7], in the dirty limit,
the superconducting gap has the same value for all points
on the Fermi surface, in other words it is isotropic. The
gap width in pure materials can show considerable
anisotropy in momentum space as shown by Richards
through infrared measurements on single crystals of Sn [8].
T. Timusk
<timusk@mcmaster.
ca>, Department of
Physics and
Astronomy, McMaster
University, Hamilton,
ON, L8S 4M1
and Member,
Quantum Materials
Program, Canadian
Institute for Advanced
Research, Toronto,
ON, M5G 1Z8
LA PHYSIQUE AU CANADA / Vol. 67, No. 2 ( avr. à juin 2011 ) C 99
THE SUPERCONDUCTING ENERGY GAP (TIMUSK)
superconductor as there is in the dirty s-wave case [12]. The
final result is that while there is a notable onset of absorption
(shown in figure 2 from the work of Hwang et al [13]) the
frequency of this feature (at 500 cm-1 in this example) is the
combined energy of the gap and the boson mode that acts as the
glue binding the superconducting carriers.
Fig. 2
(color online):The reflectance of a high temperature
superconductor with Tc = 91 K at various temperatures
from Hwang et al [13] Note the sharp onset of absorption at
27 K in the superconducting state, above a frequency of
500 cm-1. In this clean limit superconductor the onset
marks the energy where the incoming photon can break a
Cooper pair and generate a bosonic excitation.
With the discovery of high temperature superconductivity by
Bednorz and Müller [9] one of the first questions raised was the
nature of the superconducting gap. After an initial period of
confusion, it was found that the gap in these materials had
several novel properties. Early infrared spectroscopy showed
that the scattering rate was very low placing the materials in the
clean limit [10], which made the determination of the gap width
difficult. Microwave measurements by the University of
British Columbia group [11] of the penetration depth variation
with temperature are shown in figure 1. This shows
conclusively that the gap was highly anisotropic in momentum
space, going to zero for electrons travelling in certain
directions, and yielding a gap magnitude that varied as (kx2 - k2y )
around the Fermi surface. This corresponds to a d-wave
superconducting state with nodes as opposed to the isotropic swave gap of the conventional superconductors. Calculations
show that there is no onset of absorption at 2Δ in the d-wave
Another puzzle surrounding the nature of the gap of the high
temperature superconductors came from several hints that the
gap remained in the normal state at temperatures above the
superconducting transition temperature. The first evidence
from this came from NMR experiments which found a gaplike depression of the density of states at the Fermi surface
below a temperature T* which was larger than Tc at lower
doping levels but approached Tc near optimal doping. It was
initially called the "spin gap" [14]. Subsequent experiments on
the optical conductivity [15] showed that the gap involved
charge degrees of freedom and was renamed the pseudogap. A
number of experiments, including specific heat [16], confirmed
these early results [17]. Among them is angle resolved photo
emission [18,19] that showed the pseudogap had the same dwave symmetry as the superconducting gap.
The nature of the pseudogap is still under active discussion.
There are two main approaches. One view is that the
pseudogap is simply a precursor to the superconducting gap
where pairs of electrons are formed but unable to form a
coherent superconducting state due to thermal fluctuations.
The other is that the pseudogap is the signature of a completely
new quantum phase that competes with superconductivity. At
this point, it is not clear which view is correct and several
experimental groups are searching for tell-tale signatures of the
new phase.
ACKNOWLEDGEMENTS
We thank J.P. Carbotte for helpful discussions. This work has
been supported by the Natural Science and Engineering
Research Council of Canada and the Canadian Institute for
Advanced Research.
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