As identified by Perryman, et al. (1993) optical
detectors based on superconducting materials offer two important
advantages over those based on semiconductors: (a) according to the
BCS theory of superconductivity the energy gap
between
the ground state, as represented by the bound Cooper pairs, and the first
excited state, containing the broken Cooper pairs known as "quasiparticles'',
is generally more than
times smaller than the energy gap between the valence
and conduction bands of a semiconductor (Bardeen et al.
1957); (b) the Debye
energy
is much larger than the superconducting energy gap, thereby
allowing phonons created as a result of the photoabsorption process to break
Cooper pairs and create free charge (Wood & White 1969). For example in bulk
Al
while
is as large as
.
In a superconductor the conduction electrons at a particular transition
temperature interact with the lattice (an attractive electron-phonon
interaction) which overcomes their mutual Coulomb repulsion leading to the
formation of electron pairs. It is these Cooper pairs which carry the
electrical current. The temperature at which the phase transition occurs, when
electrons begin to form into condensates of pairs, is known as the critical
temperature . The Debye energy can be interpreted here as the maximum
energy associated with the vibrational modes of the lattice. Table 1 (click here) summarises
some of the key parameters of some elemental superconductors.
Table 1: Basic parameters of some elemental superconductors. is the
critical temperature,
the Debye energy, and
the energy
gap.
is the critical magnetic field above which the superconducting
state of the material is either destroyed or modified
At sufficiently low temperature (typically about an order of magnitude
lower than the superconductor's critical temperature ) the
initial number of quasiparticles
created as a result of the
absorption of a photon of wavelength
, can be in excess of
any thermally induced population and is inversely proportional to the
photon wavelength
. In general,
can be written:
where is the temperature-dependent energy gap.
The mean energy,
, required to create a single
quasiparticle in Nb and Sn has been calculated to be
and
respectively (Kurakado 1982;
Rando et
al. 1992). The variance on
defines the fundamental
limit to the intrinsic resolution
of the superconductor.
This limiting resolution, known as the Fano-limit, can be written as:
where, in both equations, the bandgap is in MeV and
is
in nm. F is the Fano factor (Fano 1947) which has been shown to be
and
for Nb and Sn respectively (Kurakado 1982;
Rando et al.\
1992). The values
and
can be considered as
typical of many of the elemental superconductors.
In superconductors such as Nb or Al operating at , with
critical temperatures of 9.2 and 1.14 K respectively, the initial charge
created by the photoabsorption of an optical photon
with
nm is
of order
and
quasiparticles respectively. The corresponding
Fano-limited wavelength resolution in Nb or Al is
nm and
nm respectively.
Equations (1) and (2) can be further generalised to any superconductor
through use of the approximate BCS relation in the weak coupling limit
which links the bandgap at absolute zero to its such that
, where k is Boltzmann's constant (for a
strongly-coupled superconductor such as Nb, a somewhat better
approximation is
) giving:
Figure 1 (click here) illustrates this fundamental limiting
spectral resolution for a number of elemental
superconductors. For
Al the resolution between Lyman-
(
and
ranges from
to 50 nm respectively, with
ranging
from
to
. If a low
superconductor
such as Hf were used, the Fano-limited resolution would, over the same
wavelength band, range from 0.2 to 15 nm.
Figure 1:
The Fano and tunnel noise-limited resolution,
and
(lower and upper sloping lines respectively),
versus wavelength for three superconducting materials. The
electronics-corrected measured resolution
(nm) for a
Nb-based symmetrical junction is also shown. The shaded
regions indicate the range of possible resolutions
to
which will depend on the design of the STJ and the role of
multiple tunnelling (Sect. 3.2)