Proceedings of the 29th International Conference on Low Temperature Physics (LT29)
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Proc. 29th Int. Conf. Low Temperature Physics (LT29)
JPS Conf. Proc. 38, 011069 (2023)
https://doi.org/10.7566/JPSCP.38.011069

Evaluation of Magnetic Field Induced by Broken
Time-Reversal Symmetry Superconductors
Mitsuhiro TESHIGAWARA1*, Yasunori MAWATARI2, Hirotake YAMAMORI2,
Rikizo YANO1, and Satoshi KASHIWAYA1
12

Department of Applied Physics, Nagoya University, Nagoya 464-8603, Japan
National Institute of Advanced Industrial Science and Technology, Tsukuba 305-8568, Japan*E-mail: teshigawara.mitsuhiro.n5@s.mail.nagoya-u.ac.jp(Received July 30, 2022)
Conventional Superconductors keep time-reversal symmetry (TRS), while several
unconventional superconductors are expected to break the TRS. The most reliable evidence
for establishing the broken TRS superconductor is the direct detection of the spontaneous
magnetic field generated by the superconductor. Since the magnetic field is weak and local,
fine-tuning the sensor's position relative to the sample is required. Here we propose aphenomenological model that can describe the spontaneous magnetic field coexisting with theMeissner effect by modifying the London equation. We numerically calculated the spatial
distribution of the spontaneous magnetic field using a finite element method and verified thata superconducting quantum interference device (SQUID) can detect the magnetic field when atiny crystal of broken TRS superconductor is placed just above the SQUID loop.
KEYWORDS: Broken Time-Reversal Symmetry Superconductor, Spontaneous
Magnetic Field, Superconducting Quantum Interference Device (SQUID)

1. Introduction
Time-reversal symmetry (TRS) has been one of the fundamental concepts incondensed matter physics. Although conventional superconductors which follow theBardeen-Cooper-Schrieffer (BCS) theory keep the TRS, several unconventional
superconductors, such as Sr2RuO4, have been reported to break the TRS [1]. The broken
TRS superconductors have been predicted to yield a spontaneous current andspontaneous magnetic field near the edge. Since the Meissner effect shields the generated
magnetic field, the spontaneous magnetic field is weak and local. One of the most reliable
tests of the TRS is to detect the spontaneous magnetic field using a micrometer-scale
magnetometer with high sensitivity.
As a candidate of the broken TRS superconductor, we assume Sr2RuO4 in thefollowing. Although the broken TRS has been suggested in SR and Kerr effectmeasurements, several previous studies performed to detect the spontaneous magnetic
field have been unsuccessful [2-7]. We believe that one of the serious issues is the lackof a method for numerically evaluating the induced field distribution to optimize theexperimental setups. In addition, the field distribution is seriously modified by theMeissner effect coexisting with the edge current, which is entirely different from that in011069-1©2023 The Author(s)This article is published by the Physical Society of Japan under the terms of the Creative Commons Attribution 4.0 License. Any further distribution of this work
must maintain attribution to the author(s) and the title of the article, journal citation, and DOI.

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JPS Conf. Proc. 38, 011069 (2023)

ordinary normal conductors. We plan to test the TRS of superconductors by detecting
the magnetic flux using the superconducting quantum interference device (SQUID). TheSQUID is a loop-shaped superconducting device with one or two Josephson junctions
and can detect magnetic flux that couples to the loop with high sensitivity. However, theoptimized setup and expected signal amplitude to detect the spontaneous magnetic field
are unclear.
Here we propose a practical phenomenological model to calculate the magnetic field
distribution around broken TRS superconductors. In previous work, the authors
calculated the magnetic field and current distribution of a 2D chiral p-wave
superconductor with the quasi-classic approximation [8]. Solving the same equation in acomplicated 3D shape is quite hard. Instead, we propose calculating the modified
London equation to reproduce the spontaneous magnetic field obtained in the previousmodel. The magnetic field distribution outside the superconductor can be obtained bysolving the Maxwell equation. Based on this model, the magnetic flux coupling to theSQUID loop is calculated in realistic experimental setups. Finally, we examine whether
we can detect the TRS breaking using the SQUID.
2. Spatial distribution of the magnetic field
2.1 Modified London equation
The London equation describes the magnetic field distribution inside thesuperconductors under the Meissner effect. Although the magnetic field source isusually applied from outside the superconductors, the current and magnetic field is
induced spontaneously in the case of the broken TRS superconductor. To introduce thiseffect phenomenologically into the equation, we added a source term that can induce a
spontaneous current. The modified London equation is given by(a)(b)

0.01Bz/BC

0-0.01
-0.02
-0.03
-0.04
05x/01015Fig. 1(a). Schematic of the semi-infinite 2D chiral p-wave superconductor. (b)Calculated magnetic field distribution along the x-axis is plotted. The blueline is the magnetic field calculated in the present work, and the red is thatin the previous work [8]. The magnetic field is normalized by criticalmagnetic field 𝐵𝑐 = 𝛷0 /2 2𝜋𝜆𝜉, where 𝛷0 is the flux quantum, 𝜆 = 3 μmand 𝜉 = 66 nm are penetration depth and coherent lengths of Sr2RuO4
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JPS Conf. Proc. 38, 011069 (2023)

1) 𝐴 = −𝐶 exp(−𝑘 2 𝑟 2 ) . #(1)𝜆2Here, A is the vector potential (B=rot A, B is the magnetic field), λ is the penetration
depth, r is the distance from the edge, and k and C are the fitting parameters. The
additional term in Eq. (1) contributes to spontaneously inducing the current andmagnetic field near the edges. In order to get the consistency of the previous work [8], kand C are set to 5.0 × 106 1/m and 8.6 × 102 T/m respectively. For materialparameters, we refer to those of Sr2RuO4 [9]. We assumed a semi-infinite 2D chiralp-wave superconductor shown in Fig. 1(a) and calculated the magnetic field distribution
as the function of distance from the edge. It is clear that the induced magnetic field in
the present model well fits the theoretical curve shown by the red line. Therefore, weconclude that the modified London equation can describe the spontaneous magnetic
field distribution around the broken TRS superconductor.
(∇2 −(a)(b)(c)

Fig. 2 Schematic of the flake of a broken TRS superconductor and a magnetic field
distribution. (a) Schematic drawing of the flake. (b) (c) Magnetic distribution at thecross sections indicated by blue and red plane in (a), respectively. The color plotshows the amplitude of the z component of the magnetic field B normalized by thecritical magnetic field 𝐵𝑐 . The arrow represent the magnetic field vector whose
length corresponds to the logarithmically scaled amplitude of the magnetic field.

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JPS Conf. Proc. 38, 011069 (2023)

2.2 Spatial distribution of magnetic field near a 3D superconducting flake
Next, we applied the modified London equation to calculate the magnetic field
distribution in 3D models. For numerical calculation, a finite element method was used[10]. We calculated the modified London equation inside the superconductor, while the(b)

(a)

(d)

(e)

 /00.020.01

0.04(f) 0.05

0.03

0.04

0.02

0.03

 /00.03

 /0(c)

0.01
00.02
0.01

-0.01

0024z (m)

680

51015x (m)

00

51015x = y (m)Fig. 3 (a-c) Setup assumed in the present calculation. As the SQUID loop, we assume
a square loop (10×10 µm2) depicted by the brown square. (a) Schematic drawing when
the superconductor moves in the direction of the z-axis (x=0, y=0).
(b) x-axis (y=0, z=0), and (c) diagonal (x=y, z=0). Here, the (x, y, z) is the center bottomof the flake. The center of the loop is at the origin of the coordinates. (d)-(f) Magnetic
flux coupling to the loop as the function of the position corresponding to the setup
shown in (a)-(c), respectively. The amplitude of the magnetic flux is normalized by theflux quantum.

Maxwell equation was used in the vacuum region. We chose a realistic shape of thesuperconducting flake expected to use for actual measurements, as shown in Fig.2(a).
Calculated results shown in Figs. 2(b) and 2(c) successfully describe the compatibilityof the spontaneous field and the Meissner effect: the appearance of a circulating current
and induced magnetic field only along the edge of the flake and the complete
suppression of the magnetic field inside the bulk. These results are consistent with what
we expect for the spontaneous field.
3. Magnetic Flux coupling to the Superconducting loop
We evaluated the magnetic flux coupling to the superconducting loop to determine
whether SQUID could detect the spontaneous magnetic field. We introduced a 2D
square loop corresponding to the superconducting loop (10×10 µm2) and evaluated themagnetic flux coupled to the superconducting loop by integrating the magnetic field
threading the loop. To find the optimum setup, we moved the superconductor in thedirection along the z-axis (Fig. 3(a)), the x-axis (Fig. 3(b)), and diagonally (x=y) (Fig.

Proceedings of the 29th International Conference on Low Temperature Physics (LT29)
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JPS Conf. Proc. 38, 011069 (2023)

011069-53(c)). The calculated amplitudes of the magnetic flux are shown in Fig. 3(d), (e), and (f),respectively.
Based on these results, the optimum position of the superconductor to detect themagnetic flux is found to be directly on top of the loop and just beside the edge. As therelative distance is increased by several micrometers, the amount of flux was reduced byorder of magnitude. The result clarify that the SQUID magnetometer directly attachedto the superconductor is more favorable than the scanning SQUID because the sensor of
the scanning SQUID is usually located more than several micrometers away from thesuperconductor.
To discuss whether the magnetic flux due to the broken TRS superconductor isdetectable, we evaluated the magnetic flux resolution of the SQUID based on the energy
resolution [11, 12]. Using the value of the designing parameters of the SQUID in ourprocess (inductance of the SQUID loop L=128.8 pH, Temperature T=4.2 K, coupling
constant between the input coil and the SQUID loop k = 0.66), the magnetic flux
resolution of 0.8μ𝛷0 Wb/ Hz is obtained. On the other hand, the flux coupling to theloop is 0.04Φ0 when we place the superconductor flake in the middle of the SQUIDloop. Since the spontaneous magnetic field is static, we do not need a wide-frequency
operation of the SQUID. Therefore, our SQUID has enough resolution to detect themagnetic flux from the superconductor and the present measurement method is valid fortesting the TRS of the superconductor. Although the edge current reflects the nontrivial
topology of chiral p-wave superconductivity, the effect of the surface roughness on thespontaneous magnetic field should also be considered. Since the reduction of the field issmall as shown in Ref. [13], the present results are valid even when the surface
condition is far from the specular limit.
3. Conclusion
We proposed a practical measurement method to test the TRS of the superconductor
by detecting the spontaneous magnetic field using a SQUID. To calculate the spatial
distribution of the spontaneous field, we introduced a source term into the London
equation and calculated the magnetic flux in 3D space using a finite element method.
We found the optimum position of the superconductor to detect the magnetic flux
efficiently. Since the calculated amplitude of the magnetic flux is large enough to bedetected by the SQUID, the TRS breaking can be tested by the SQUID. In future work,
the magnetic field distribution around a flake with a complicated structure will bediscussed. Also, the influence of trapped flux is evaluated to exclude the extrinsic
origins of the magnetic field generation. We expect the present model provides avaluable guide to checking the validity of theoretical calculation by comparing
experimental data and numerical calculation.

Acknowledgment
This study was supported by KAKENHI (Grant Nos. 18H01243, 20H00131,20K05314, 21H01009, 21H04652 and 21K13854).

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