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Superconducting Stiffness and Coherence Length of FeSe_{0.5}Te_{0.5} Measured in a Zero-Applied Field

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## Abstract

**:**

## 1. Introduction

## 2. Experimental Setup

## 3. Measurements

#### 3.1. Stiffness and Critical Current

#### 3.2. Susceptibility

#### 3.3. Hysteresis

#### 3.4. Critical Magnetic Fields

## 4. Analysis Model

#### 4.1. Stiffness

#### 4.2. Coherence Length

## 5. Data Analysis

## 6. Reproducibility and Origin of the Knee

## 7. Discussion

#### 7.1. The Knee

#### 7.2. Critical Exponents

#### 7.3. First Critical Field

## 8. Conclusions

## Author Contributions

## Funding

## Data Availability Statement

## Acknowledgments

## Conflicts of Interest

## Appendix A. Temperature Calibration

**Figure A1.**Temperature Calibration. Temperature dependence of the magnetic moment of a disconnected FST ring in the presence of a magnetic field, repeated for different ${I}_{\mathrm{ec}}$, as indicated by the colors. (

**a**) Before calibration. (

**b**) After calibration.

**Figure A2.**The influence of the leaking field from the excitation coil on the measurements. (

**a**) Calibrated measurements in the presence of positive and negative current values, as indicated by the colors. (

**b**) Averaging over the directions of the currents in (

**a**).

**Figure A3.**Critical currents before and after the calibration. Critical current vs. the temperature before the calibration in gray diamonds and after in blue circles. The inset shows the temperature correction $\mathsf{\Delta}T$ vs. the current in the excitation coil. The relation is approximately parabolic.

**Figure A4.**Estimation of the temperature calibration process errors. Measurements after temperature calibration (from Figure A1b) without current in the excitation coil in black circles and with a current of ${I}_{\mathrm{ec}}=25$ mA in red circles. The error is estimated by the temperature difference between two points with the same moment value. It is represented by $\delta T$ and depends on the current and the temperature.

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**Figure 1.**Experimental setup: (

**a**) A scanning electron microscope image of a single crystal of FST, from which the ring was cut out. (

**b**) A microscopic image of the Fe${}_{1+y}$Se${}_{0.5}$Te${}_{0.5}$ ring. The sample is not uniform. The minimal height, inner, and minimal outer radii are $h=0.10$ mm, ${r}_{\mathrm{in}}=0.26$ mm, and ${r}_{\mathrm{out}}=0.50$ mm, respectively. (

**c**) A copper excitation coil and a superconducting ring beside it. The coil has a length of 60 mm, an outer diameter of 0.25 mm, and 9300 turns in two layers. (

**d**) The ring and excitation coil assembly moves rigidly relative to a gradiometer, connected to a SQUID system (not shown), and surrounded by a main coil for field zeroing or field-dependent measurement. The SQUID, gradiometer, and main coil are part of a QD-MPMS3 system.

**Figure 2.**Data: (

**a**) Stiffness measurements. SC’s magnetic moment vs. the current in the excitation coil at different temperatures, indicated by the colors. The inset is focused on the measurement at 12 K. A linear relation is found for low currents. At some critical current value, the signal drops to zero. The blue circles in (

**b**) depict the temperature dependence of the linear slope obtained at low currents (far from ${I}_{\mathrm{ec}}^{c}$) in panel (

**a**). (

**b**) Critical temperature. SC’s current in the EC vs. the temperature (red down-pointing triangles), as described in Section 3.1; measured susceptibility (with a minus sign) vs. the temperature (emerald diamonds) in MKS units in the presence of a magnetic field of $\phantom{\rule{3.33333pt}{0ex}}1$ mT and without an excitation coil (according to Section 3.2). Inset (

**b**) shows the critical currents vs. the calibrated temperature (extracted from the breakpoints in panel (

**a**)).

**Figure 3.**Magnetic measurements: (

**a**) Magnetic hysteresis loop above the critical temperature. (

**b**) $m\left(H\right)$ at different temperatures below ${T}_{c}$, as indicated by the colors. Inset: The temperature dependence of the critical fields ${H}_{c1}$ (blue circles) and ${H}_{c2}$ (red down-pointing triangles) on the left and right Y-axis, respectively.

**Figure 4.**Penetration depth and coherence length. (

**a**) The right Y-axis shows the penetration depth as a function of the temperature in blue and emerald triangles, for the different calibration methods. The left Y-axis shows the temperature dependence of the coherence length. The red-circles are taken from the critical current measurement in Figure 2b-inset through Equation (14) with the measured $\lambda $, and the black-squares are from the second critical field in Figure 3b-inset with Equation (17). Panels (

**b**,

**c**) are log-log plots of the stiffness ${\lambda}^{-2}$ and $1/\xi $ vs. $1-T/{T}_{c}$, respectively. The linear regression represents the critical exponents according to Equations (15) and (16), respectively. Earlier stiffness measurements using the $\mathsf{\mu}$SR method have been added to (

**b**) in brown [3] and yellow [4] stars. The same power law is fitted to this data. For comparison, we add to (

**c**) asterisks reflecting the measurements of $1/\xi $ from the resistivity method [10] in magenta, ARPES [11] in purple, and STM [9] in green.

**Figure 5.**Reproducibility. Normalized magnetic moment $m/m(T\to 0)$ vs. temperature for different rings. (

**a**) in the presence of current in the excitation coil, as described in Section 3.1. (

**b**) in the presence of an applied field perpendicular to the ring. The central ring of this research is 1. An offset is added for clarity.

**Figure 6.**The knee’s field dependence. Temperature dependence of the difference between the ring’s moment measured with the excitation coil current and field, to a measurement with the field only (blue circles, left Y-axis). The brown circles on the right Y-axis are the measurement in the zero-field current in the excitation coil. The data are shifted for clarity. The knee is at the same temperature regardless of the field. The inset shows the ring’s moment vs. the temperature in the presence and absence of an applied field (1 mT) and current in the EC (10 mA). The data in the main panel are obtained by subtracting the two datasets in the inset.

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**MDPI and ACS Style**

Peri, A.; Mangel, I.; Keren, A.
Superconducting Stiffness and Coherence Length of FeSe_{0.5}Te_{0.5} Measured in a Zero-Applied Field. *Condens. Matter* **2023**, *8*, 39.
https://doi.org/10.3390/condmat8020039

**AMA Style**

Peri A, Mangel I, Keren A.
Superconducting Stiffness and Coherence Length of FeSe_{0.5}Te_{0.5} Measured in a Zero-Applied Field. *Condensed Matter*. 2023; 8(2):39.
https://doi.org/10.3390/condmat8020039

**Chicago/Turabian Style**

Peri, Amotz, Itay Mangel, and Amit Keren.
2023. "Superconducting Stiffness and Coherence Length of FeSe_{0.5}Te_{0.5} Measured in a Zero-Applied Field" *Condensed Matter* 8, no. 2: 39.
https://doi.org/10.3390/condmat8020039