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2. The photoabsorption process in superconductors

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 tex2html_wrap_inline1027 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 tex2html_wrap_inline1029 times smaller than the energy gap between the valence and conduction bands of a semiconductor (Bardeen et al. 1957); (b) the Debye energy tex2html_wrap_inline1031 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 tex2html_wrap_inline1033 while tex2html_wrap_inline1035 is as large as tex2html_wrap_inline1037.

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 tex2html_wrap_inline1039. 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.

  table217
Table 1: Basic parameters of some elemental superconductors. tex2html_wrap_inline1041 is the critical temperature, tex2html_wrap_inline1043 the Debye energy, and tex2html_wrap_inline1045 the energy gap. tex2html_wrap_inline1047 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 tex2html_wrap_inline1051) the initial number of quasiparticles tex2html_wrap_inline1053 created as a result of the absorption of a photon of wavelength tex2html_wrap_inline1055, can be in excess of any thermally induced population and is inversely proportional to the photon wavelength tex2html_wrap_inline1057. In general, tex2html_wrap_inline1059 can be written:
equation236
where tex2html_wrap_inline1061 is the temperature-dependent energy gap. The mean energy, tex2html_wrap_inline1063, required to create a single quasiparticle in Nb and Sn has been calculated to be tex2html_wrap_inline1065 and tex2html_wrap_inline1067 respectively (Kurakado 1982; Rando et al. 1992). The variance on tex2html_wrap_inline1069 defines the fundamental limit to the intrinsic resolution tex2html_wrap_inline1071 of the superconductor. This limiting resolution, known as the Fano-limit, can be written as:
equation241
where, in both equations, the bandgap tex2html_wrap_inline1073 is in MeV and tex2html_wrap_inline1075 is in nm. F is the Fano factor (Fano 1947) which has been shown to be tex2html_wrap_inline1079 and tex2html_wrap_inline1081 for Nb and Sn respectively (Kurakado 1982; Rando et al.\ 1992). The values tex2html_wrap_inline1083 and tex2html_wrap_inline1085 can be considered as typical of many of the elemental superconductors.

In superconductors such as Nb or Al operating at tex2html_wrap_inline1087, with critical temperatures of 9.2 and 1.14 K respectively, the initial charge tex2html_wrap_inline1089 created by the photoabsorption of an optical photon with tex2html_wrap_inline1091 nm is of order tex2html_wrap_inline1093 and tex2html_wrap_inline1095 quasiparticles respectively. The corresponding Fano-limited wavelength resolution in Nb or Al is tex2html_wrap_inline1097 nm and tex2html_wrap_inline1099 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 tex2html_wrap_inline1101 such that tex2html_wrap_inline1103, where k is Boltzmann's constant (for a strongly-coupled superconductor such as Nb, a somewhat better approximation is tex2html_wrap_inline1107) giving:
eqnarray254
Figure 1 (click here) illustrates this fundamental limiting spectral resolution tex2html_wrap_inline1109 for a number of elemental superconductors. For Al the resolution between Lyman-tex2html_wrap_inline1111 (tex2html_wrap_inline1113 and tex2html_wrap_inline1115 ranges from tex2html_wrap_inline1117 to 50 nm respectively, with tex2html_wrap_inline1119 ranging from tex2html_wrap_inline1121 to tex2html_wrap_inline1123. If a low tex2html_wrap_inline1125 superconductor such as Hf were used, the Fano-limited resolution would, over the same wavelength band, range from 0.2 to 15 nm.

  figure270
Figure 1: The Fano and tunnel noise-limited resolution, tex2html_wrap_inline1127 and tex2html_wrap_inline1129 (lower and upper sloping lines respectively), versus wavelength for three superconducting materials. The electronics-corrected measured resolution tex2html_wrap_inline1131 (nm) for a Nb-based symmetrical junction is also shown. The shaded regions indicate the range of possible resolutions tex2html_wrap_inline1133 to tex2html_wrap_inline1135 which will depend on the design of the STJ and the role of multiple tunnelling (Sect. 3.2)


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