# Heating Powers

The flow of the cooling current through the SSmS junction sets up a number of heating powers, the more prominent of which are described below. These heating and cooling power flows eventually reach an equilibrium with the junction settling at a equilibrium temperature for a given voltage.

## Electron-Phonon coupling

Electron-phonon coupling results in a heat flow between the electron gas and the lattice as the former is cooled. This results in a drain on the cooling power as a proportion is essentially wasted by cooling the lattice. This drain is modelled by the equation [1,2]

with Σ being the material specic coupling constant, λ the volume of the absorber, and T(_{sm,b}) the electron and phonon temperatures respectively.

## Ohmic heating

Ohmic heating takes place within the semiconductor as a direct result of the current, I, owing through it. This is given by the standard Ohmic heating equation, with R_{sm} being the resistance of the semiconductor used.

## Quasi-Particle effects

The fow of the cooling current through the tunnel junction leads to the creation of quasiparticles [3] in the superconductor. If they do not diffuse out of the tunnelling region quickly, these quasiparticles are capable of returning heat to the semiconductor electrode via two mechanisms.

**Recombination**

There is a finite chance that two quasiparticles within the superconductor will recombine to produce a phonon which can then be absorbed by the semiconductor.

**Back-Tunnelling**

The second mechanism involves the probability that quasiparticles may tunnel back through the barrier into the semiconductor, eectively decreasing the net cooling current across the junction [4].

These two mechanisms can be modelled jointly by the term [5]

,

a simplifying assumption where β < 1 denotes the fraction of the power deposited in the superconducting electrode that is returned to the semiconductor. β is a parameter dependant only on the temperature of the surrounding bath, with

being the net power dissipated in the superconductor.

## Density of States Broadening

A dirtying parameter, Γ, has been proposed [6,7] to describe the availability of states in the superconductor band gap. This requires the superconductor density of states equation to be re-written as

in which Γ manifests itself as a broadening factor as shown below. This enables electrons below the threshold energy of (E-eV) to tunnel out of the semiconductor, reducing the average energy removed from the system per electron. This effect presents itself as a loss of filtration which results in a net heating power.

Figure 6: Increasing Γ raises the valley height of the density of states, allowing electrons to occupy states in the previously forbidden region.

## References

- M. Leivo et al., Applied Superconductivity
**5**(1998). - M. Roukes et al., Physical Review Letters
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**83**(1998). - P. Fisher et al., Applied Physics Letters
**74**(1999). - R. Dynes et al., Physicsal Review Letters
**53**(1984). - B. Mitrovic and L. Rozema, Journal of Physics: Condensed Matter
**20**(2008).