8.6 Qualitative model of inhomogeneity in a voltage-biased self-heating hotspot
Table 8.4.
Sample
ρ (μΩcm)
TC,AL (K)
ΔTC (K)
D (cm2 /s)
B20 C20 D20
407 294 180
2.0 2.5 3.05
0.09 0.06 0.03
0.62 0.64 0.77
2
with L = π3 ( keb )2 the Lorenz number, TC the critical temperature, ρ the resistivity, and D the diffusion constant. The electron-phonon scattering time is estimated using the theory for electronphonon scattering in dirty metals of A. Schmid [30]: τe,ph ≈
2 3 ωD (kB T )3 B
(8.11)
with ωD the Debye frequency and B a parameter describing the effect of the short mean free path in dirty metals. The Debye temperature for TiN ΘD ≈ 580 K [31]. For our films the disorder parameter kF l ≈ 5, with kF the Fermi wave vector and l the elastic mean free path [7]. At the critical temperature, following the work of Schmid, this leads to a B ≈ 10. Table 8.5 shows the electron phonon scattering time, thermal healing length, minimum current for sustaining a stable normal region, determined using Eqs. 8.11, 8.9, and 8.5 and 8.7 respectively. The input parameters used, as previously measured in sections 8.2, 8.3, 8.4, are repeated in Table 8.4. The last column of Table 8.5 gives the bandwidth over which the minimum stable current varies (ΔIm ), given the width of the resistive transition (ΔTC ) determined in Section 8.3. The relatively large difference in minimum stable current between the different wires, originates from the difference in critical temperature, resistivity, and thickness of the wires. The minimum stable currents given in Table 8.5 are calculated for segments at the ends of the normal region of the wire, with a contact at√the bath temperature. From Eq. 8.7 it follows that this current is a factor of 2 larger than the minimum stable current in the center of the wire. The bandwidth over which the minimum stable current varies (ΔIm ) is always significantly smaller than the difference between the minimum current with cooling at the contacts Im and the minimum current with heat flow only to the phonons I1 . Therefore, we expect the normal region to shrink from the contacts under decreasing bias, even though our wires may contain electronic inhomogeneities. Fig. 8.13 shows the simulated voltage-biased current-voltage characteristics for decreasing voltage-bias, going from a biased wire in the normal state to a wire completely in the superconducting state. The curve is simulated by 121