# 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]

$P_{e-ph}=\Sigma\lambda\left(T^{5}_{sm}-T^{5}_{b}\right)$

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 Rsm being the resistance of the semiconductor used.

$P_{ohm}-I^2R_{sm}$

## 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]

$\beta P_{s}$,

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

$P_{s}=IV+P_{cool}$

being the net power dissipated in the superconductor.

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

$g(E)=Real\left|\frac{E-i\Gamma}{\sqrt{(E-i\Gamma)^2-\Delta^2}}\right|$

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

1. M. Leivo et al., Applied Superconductivity 5 (1998).
2. M. Roukes et al., Physical Review Letters 55 (1985).
3. E. Kaxiras, Atomic and Electronic Structure of Solids, Cambridge University Press, 2003.
4. J. Jochum et al., Journal of Applied Physics 83 (1998).
5. P. Fisher et al., Applied Physics Letters 74 (1999).
6. R. Dynes et al., Physicsal Review Letters 53 (1984).
7. B. Mitrovic and L. Rozema, Journal of Physics: Condensed Matter 20 (2008).