We proposed a new high resolution single photon infrared spectrometer for search for radiative decay of cosmic neutrino background(C$ u$B). The superconducting-tunnel-junctions(STJs) are used as a single photoncounting device. Each STJ consists of Nb/Al/Al${}_{mathrm{x}}$O${}_{mathrm{y}}$/Al/Nb layers and their thicknesses are optimized for the operation temperature at 370 mK cooled by a ${}^{3}$He sorption refrigerator. Our STJs achieved the leak current 250 pA and the measured data implies that a smaller area STJ fulfills our requirement. FD-SOI MOSFETs are employed to amplify the STJ signal current in order to increase signal-to-noise ratio(S/N). FD-SOI MOSFETs can be operated at cryogenic temperature of 370 mK, which reduces the noise of the signal amplification system. FD-SOI MOSFET characteristics are measured at cryogenic temperature. The Id-Vgs curve shows a sharper turn on with a higher threshold voltage and the Id-Vds curve shows a non linear shape in linear region at cryogenic temperature. Taking into account these effects, FD-SOI MOSFETs are available for read-out circuit of STJ detectors. The bias voltage for STJ detectors are 0.4 mV and it must be well stabilized to deliver high performance. We proposed an FD-SOI MOSFET based charge integrated amplifier design as a read-out circuit of STJ detectors. The requirements for an operational amplifier used in the amplifier is estimated using SPICE simulation. The op-amp required to have a fast response(GBW$geq$100 MHz) and it must have low power dissipation as compared to the cooling power of refrigerator.
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