Transition to the normal state induced by high current densities in ${\text{YBa}}_2{\text{Cu}}_3{\text{O}}_{7-\delta }$ thin films: A thermal runaway account

By using a finite-element method we analyze at a quantitative level the abrupt jump to the normal state in high-$T_c$ films observed when measuring their current-voltage characteristics at current densities, $J^{\ast }$, which are between two to three times their critical current density, $J_c$. The experimental data that this analysis focuses on are from ${\text{YBa}}_2{\text{Cu}}_3{\text{O}}_{7-\delta }$ films, measured between 75 K and $T_c\approx 90\text{K}$ and under zero applied magnetic field. Our main starting point is the assumption that the constant-temperature curves, i.e., that would be measured at arbitrarily short measuring time, are smooth and so jumpless. When taking into account the finite measuring times, the highly nonlinear nature of the film’s electrical conductivity, and the thermal properties of the substrate, simulation by the finite-element method shows that a thermal runaway takes place that explains, without free parameters, the experimental jumps to a 5% accuracy. The voltage values prior to the jump are also coherently accounted for with similar accuracy. No critical mechanism such as the vortex instability model from Larkin-Ovchinikov or any others are needed for this quantitative agreement to our measurements, though they can become dominant under different refrigeration conditions, temperature range, or magnetic-field application.

Figure 1

Experimental building procedure of the current-voltage characteristics studied in this work. A staircase ramp of current was applied to the ${\text{YBa}}_2{\text{Cu}}_3{\text{O}}_{7-\delta }$ microbridges. Each current step was applied for 1 ms. Voltage data were subsequently acquired at a rate of ${10}^5$ samples/s resulting in a slightly rounded voltage step. 50 current steps were applied in each run with a current increase between steps from 2 to 10 mA.

Figure 2

Experimental data for one of the ${\text{YBa}}_2{\text{Cu}}_3{\text{O}}_{7-\delta }$ microbridges whose $T_c=89.9\text{K}$. Six bath temperatures are drawn. More series were measured but are not shown for the sake of clarity. No data points above the jump are shown because the factor of 50 discontinuities in voltage would downplay the most relevant lower data points.

Figure 3

Low-voltage range for the same experimental data as Fig. 2. The solid lines are the constant-temperature CVCs we will be using along this paper as the background CVCs (for this sample), obtained from the fit of the functional form in Eq. (1) over the indicated voltage region (below the dotted line). These isothermal CVCs are smooth all along the current range up to the normal state.

Figure 4

Illustration of the meshing build for one of the ${\text{YBa}}_2{\text{Cu}}_3{\text{O}}_{7-\delta }$ thin films. Note that by taking advantage of the left-right symmetry, only the right half of the film is modeled. Because of the tiny size of the film with respect to the substrate an inhomogeneous meshing was needed to save degrees of freedom.

Figure 5

Block diagram illustrating the main steps in the FEM thermal analysis of the ${\text{YBa}}_2{\text{Cu}}_3{\text{O}}_{7-\delta }$ film.

Figure 6

FEM simulation results for the YBCO film’s temperature evolution. The three curves show the film’s differential temperature (relative to that of the bath) for three initials bath’s temperatures as a function of current density. The important result is the appearance of a numerical instability in the film’s temperature slightly above the bath’s temperature. The jump in temperature in these simulations is of tens of kelvins, leading the sample to the normal state.

Figure 7

Comparison between measurements of the current-voltage curves (diamond symbols) and the calculations based only on thermal effects sketched in Fig, 5 (solid lines). These data correspond to the bath temperature $T=87.4\text{K}$. Current is applied in the simulations by staircase ramps of 1 ms duration per step to match experimental conditions. The agreement is good, giving in particular a 1.5% deviation for $J^{\ast }$ and around of 16% for $E^{\ast }$. The inset shows an overview of the same comparison, here including the landing into the normal state. Note the huge discontinuity in the electric field.

Figure 8

Detailed view of one of the CVC of our $10\text{−}\mu \text{m}$-wide film showing the drift in temperature as current is fed into the sample. The film crosses successive isotherms (three of them are illustrated) until thermal instability triggers the voltage jump.

Figure 9

Experimental data of the instability current $J^{\ast }$ for the whole range of measuring temperatures of a $10\mu \text{m}$ width YBCO film. The solid line is the thermal-only FEM simulations from the jump-free CVCs of Fig. 3. The calculation error is approximately equal to the data points’ size.

Figure 10

Comparison of the voltage jump for two YBCO films of quite different width but otherwise the same normal-state resistivity. Their critical temperatures are 88.15 K ($100\mu \text{m}$ film) and 87.20 K ($10\mu \text{m}$ film). The distinct jump current is accounted for by the same thermal-only FEM simulation. The common bath temperature is 78.1 K.