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###

RSJ Model Temperature Dependence

When the junction is not operating at the nominal temperature as
specified with the `tnom` parameter, a correction factor is
applied to the critical current to account for the temperature
difference. The superconducting energy gap and therefor the junction
gap voltage will also change as a function of temperature.

If the gap voltage is not specified with the `vg` parameter or its
alias `vgap`, the model will compute the gap voltage from BCS
theory using the superconducting transition temperature (`tc`
parameter) and Debye temperature (`tdebye` parameter) at the
operating temperature.

When a gap voltage is supplied, it is taken as the value at the
nominal temperature. The model utilizes an interpolation formula to
represent temperature variation of the gap parameter of the
superconductor in the case.

In either case, the RSJ model currently assumes that the same
material appears on either side of the barrier.

Below is the approximation formula for the superconducting energy gap
that is employed when the nominal junction gap voltage is given
explicitly. The junction vap voltage is twice the superconductor
energy gap.

= *tanh*(*tcfct**tc*/*T* - 1))

Reference: `https://physics.stackexchange.com/questions/192416/
`

interpolation-formula-for-bcs-superconducting-gap
Here,
is the gap parameter, with the subscript indicating the
value at temperature T=0. The `tcfct` parameter is
semi-empirical, commonly cited as 1.74, which is the default value.

In order to apply temperature correction one takes the assumption that
parameters for the reference junction are measured at nominal
temperature `tnom`, and we therefor have

= *tanh*(*tcfct**tc*/*tnom* - 1))

The correction factor, which will multiply device gap voltage, is the
ratio
/
. This will apply when the nominal gap
voltage is given. When not given, the gap voltage is computed
directly with BCS theory and the approximation formula is not used.

The critical current temperature variation again follows BCS theory.
The temperature correction factor *tcf* multiplies the instance
critical current to account for the temperature difference between
operating and nominal temperatures.

*tcf* = (*Vg*/*Vg*_{nom})(*tanh*(*eVg*/4*KT*)/*tanh*(*eVg*_{nom}/4*KT*_{nom}))

Here, *Vg*
is the computed gap voltage, *e*
is the electron charge,
*K*
is Boltzmann's constant, *T*
is temperature in Kelvin,

** Next:** Josephson Tunnel Junction Model
** Up:** Josephson Junction Model
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Stephen R. Whiteley
2022-09-18