Author Correction: Magnetic flux trapping in hydrogen-rich high-temperature superconductors

Correction to: Nature Physics https://doi.org/10.1038/s41567-023-02089-1, published online 15 June 2023.

In the version of this article originally published, in Fig. 4c and for text in the section “Pinning and thermally activated motion of vortices,” the flux creep rate, $S$, was estimated by fitting the experimental time-dependent trapped magnetic moment, ${m}_{{trap}}\left(t\right)$, to the equation

$$S=-\frac{1}{{m}_{{trap},0}}\frac{d{m}_{{trap}}\left(t\right)}{{dln}\left(t\right)}$$

where ${m}_{{trap},0}$ refers to ${m}_{{trap}}\left(t={t}_{{delay}}\right)$, with ${t}_{{delay}}$ being the time at which measurements began. In flux creep experiments, time, $t$, is counted from the moment when the applied magnetization field, ${H}_{M}$, is reduced to zero. The ${t}_{{delay}}$ can vary widely between different flux creep experiments¹, ranging from hundreds of microseconds² to several hours³. However, the delay time cannot be zero, ${t}_{{delay}}\ne 0{s}$, due to the inherent delay between the end of applied field change and the start of measurements. To avoid misinterpretation, we rewrite the equation as

$$S=\frac{1}{\left|{m}_{{trap},{t}_{{delay}}}\right|}\left|\frac{d{m}_{{trap}}\left(t\right)}{{dln}\left(t\right)}\right|$$

where $S$ is defined to be a positive value^2,4,5.

We have replaced the original text from the start of the section up to but not including the sentence that begins “The scale of fluctuations responsible…” with the following:

“The kinetics of m_trap generated at μ₀H_M = 1 T and T_M = 165 K demonstrate an extremely low decay rate of the trapped magnetic moment in Im-3m-H₃S, even at temperatures in vicinity of T_c, where thermal fluctuations are expected to be significantly higher (see Fig. 4c). The estimated decay of m_trap is approximately 2.8% within 53.6 hours at 165 K, 3.0% within 24.3 hours at 180 K, and 4.4% within 28.4 hours at 185 K. This is comparable to the low values of dissipation of trapped magnetic flux measured in type-II superconductors⁵⁴, including the extremely slow creep rate in high-j_c MgB₂ films at substantially lower T/T_c < 0.5 (ref. 55).”

In the article, we aimed to demonstrate that the long-term flux creep decay rate in H₃S is very low and comparable to that in well-studied superconductors such as MgB₂. Our experiments followed a protocol used in applied superconductivity³, where the delay time in long-term flux creep studies is several hours (for example, ${t}_{{delay}}=5\,h$, see ref. ³), and the magnetization field is ${{\mu }_{0}H}_{M}=0.5-1.0{T}$ (see refs. ^3,6).

Therefore, in the article, we magnetized the H₃S at T = 165 K with an applied magnetic field ${{\mu }_{0}H}_{M}=1{T}$, and implemented a delay time of ${t}_{{delay}}=38\,h$. The measurements of the ${m}_{{trap}}\left(38\,h\le t\le 91\,h,T=165\,{K}\right)$ are shown in Fig. 4c (black data points). It is practically impossible to recognize any trapped flux decay over the time of experiment, ${t}_{{delay}}=38\,h\le t\le 92\,h$, by visual inspection. This practically decay-free ${m}_{{trap}}\left(38\,h\le t\le 92\,h,T=165{K}\right)$ in H₃S is similar to the practical decay-free ${m}_{{trap}}\left(5\,h\le t\le 30\,h,T=20\,{K}\right)$ of MgB₂ (see figure 8 in ref. ³) and the practical decay-free ${B}_{{trap}}\left(0.15\,h\le t\le 83\,h,T=5{K}\right)$ of Ba_0.6K_0.4Fe₂As₂ (see figure 8 in ref. ⁷), where trapped flux decay is also imperceptible by visual inspection.

Although the long-term flux creep rate, $S$, in H₃S can be estimated from our ${m}_{{trap}}\left(38\,h\le t\le 92\,h,T=165{K}\right)$ dataset, we chose to present only the experimental ${m}_{{trap}}\left(t\right)$ data in Fig. 4c to highlight the extremely low trapped flux dissipation as it was done for MgB₂ in ref. ³. This decision was made because accurately deducing the long-term flux creep rate, $S$, requires ${m}_{{trap}}\left(t,T=165\,{K}\right)$ measurements over a much longer period (estimated as $38\,h\le t\lesssim 1000\,h$ at $T=165\,{K}$), which is impractical given the allocated SQUID time within our ongoing workflow.

We observed a similar nearly dissipation-free state in H₃S when the sample was subsequently warmed up from 165 K to 180 K, and then from 180 K to 185 K after each run of creep measurements. The corresponding datasets of ${m}_{{trap}}\left(95\,h\le t\le 119\,h,T=180\,{K}\right)$ and ${m}_{{trap}}\left(133\,h\le t\le 161\,h,T=185\,{K}\right)$ are shown in Fig. 4c and we have removed the guide lines for S (see Fig. 1 below).

Fig. 1

Original and revised Fig. 4c.

We have updated the caption to Fig 4c as follows:

“Time dependent trapped magnetic moment in H₃S at several temperatures near T_c. The trapped magnetic flux was generated at μ₀H_M = 1 T and T_M = 165 K. The dataset at 165 K was measured after 38 hours after the applied magnetic field was switched off. Measurements of the magnetic flux creep at 180 K was started at 95 hours after removing of μ₀H_M and subsequent warming from 165 K. Measurements of the magnetic flux creep at 185 K was started at 133 hours after removing of μ₀H_M and subsequent warming from 180 K. The error bars represent the standard deviation from multiple measurements.”

We thank Nobuyuki Zen for suggesting that we provide a more detailed description of our flux creep experiments and for the pointing out the potential misinterpretation of the ${m}_{{trap},0}$ designation. The text and figure have been amended in the HTML and PDF versions of the article.