Phase transition in the cuprates from a magnetic-field-free stiffness meter viewpoint

A method to measure the superconducting (SC) stiffness tensor \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\bar \rho _{\mathrm{s}}$$\end{document}ρ¯s, without subjecting the sample to external magnetic field, is applied to La1.875Sr0.125CuO4. The method is based on the London equation \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\mathbf{J}} = - {\bar{\mathbf{\rho }}}_{\mathrm{s}}{\mathbf{A}}$$\end{document}J=-ρ¯sA, where J is the current density and A is the vector potential which is applied in the SC state. Using rotor free A and measuring J via the magnetic moment of superconducting rings, \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\bar \rho _{\mathrm{s}}$$\end{document}ρ¯s at T → Tc is extracted. The technique is sensitive to very small stiffnesses (penetration depths on the order of a few millimeters). The method is applied to two different rings: one with the current running only in the CuO2 planes, and another where the current must cross planes. We find different transition temperatures for the two rings, namely, there is a temperature range with two-dimensional stiffness. Additional low energy muon spin rotation measurements on the same sample determine the stiffness anisotropy at T < Tc.


SUPPLEMENTARY NOTE 1: STIFFNESSOMETER Zero External Field
There is a risk that field generated in the inner-coil leaks since no coil is infinitely long or perfect. To overcome this leak, a main coil, also shown in Fig. 2 in the manuscript, acts as a shim to cancel the field on the ring when it is at the gradiometer center. Our main-coil has a field resolution of 10 −3 Oe from 0 up to 200 Oe. Therefore, we can keep the field on the ring as low as 1 mOe.
To ensure that our signal is not due to residual field, we measure the stiffness (i.e. zero   4

Validity of London's Equation
In the Ginzburg-Landau theory, J = ρ s ( c/q∇ϕ − A) where ϕ is the phase of the order parameter and q is the carriers charge. However, as explained in the main text, the procedure of cooling in zero vector potential and turning it on only at base temperature, sets ∇ϕ = 0, and consequently J = −ρ s A. To see this one can view the phase ϕ as an in-plane arrow.
Cooling at A = 0 must set ∇ϕ = 0 to minimize the kinetic energy, namely, all the arrows point in the same direction. Since the phase is quantized, to change ϕ means to make a twist of all arrows in a closed loop, such that the phase between the first arrow and last one in the loop changes by 2π. This would lead to a discontinuity in the phase value, a procedure that costs energy. A nice analog is a ferromagnetic ring with the spins pointing in the same direction. Rotating the last spin with respect to the first one by 2π requires to break a bond. This procedure is not energetically favorable for a ferromagnet (or the SC ring). Therefore, when turning A on after cooling, all the arrows continue to point in the same direction and ∇ϕ = 0, until A exceeds a critical value. At this point, the current in the SC is too high and it is worthwhile for the superconductor to "break a bond" and reduce the current.
To prove London's proportionality in our system, we measure ∆V max R as a function of the current in the inner-coil at constant temperature. This is depicted in the inset of Fig. 3(a) in the main text for LSCO c-ring at T = 29.92(5) K and for the a-ring at T = 29.28(5).
The signal from the ring is proportional to the applied current in the inner-coil until ∆V max R reaches a saturation value. It means that the superconductor can generate only a finite amount of current. The critical vector potential allows one to determine the coherence length ξ. Using relation 4.37 from [1], Supplementary Fig. 3, and the critical current presented in Fig. 3(a) we estimate ξ ≈ 50 nm. A more accurate determination of ξ is given in Ref. [2].
We find that ξ λ at least up to T = 29.92(5) K.

SUPPLEMENTARY NOTE 2: NUMERICAL METHODS
Here we provide more details about the numeric solution of Eq. 3 in the anisotropic case.
The gauge choices are as follows: Inside the ring, applying divergence to Eq. 3 yields the  Supplementary Fig. 4 shows the numerical results of the vector potentials ratio that appears in Eq. 2 as a function of (a) (R/λ ab ) 2 and (b) (R/λ c ) 2 for λ ab = 13.9 µm at T = 29.16 K. In our analysis, λ ab is extracted from the c-ring data in the isotropic case.
Then, for each temperature, the corresponding λ ab is used to generate the result in panel 6 (b), and combining with the a-ring data λ c is extracted. Finally, Supplementary Fig. 7 shows the magnetic field generated by the ring as calculated from the curl of A R . The penetration pattern of the field is of an ellipse due to the penetration depths anisotropy.

SUPPLEMENTARY NOTE 3: LE-µSR
There are two methods by which one can extract the penetration depth. The simple method is to fit each data set (at each temperature and energy) to A(t) = A 0 exp(−t/T 2 ) cos(ωt).
From this fit one can extract asymmetry, relaxation, and the average internal field as a function of average implantation depth and temperature. Supplementary Fig. 8 summarizes the internal magnetic field as a function of implantation energy for different temperatures and field orientations. The field here is calculated by B = ω/2πγ, where ω is the angular frequency of the muon polarization and γ is the gyromagnetic ratio. Noticeably, close to the surface and at low T , the magnetic field does not change with increasing implantation depth for H c. Only for energies above 5 keV does a linear trend of decay appears. This 10 to 20 nanometers of "dead layer" could be a byproduct of the polishing process.
Supplementary Fig. 9 depicts the temperature dependence of the individual fit parameters for the highest implantation energy. The magnetic field (panel (a)) seems to behave erratically close to the phase transition into the superconducting state. We attribute this behavior to demagnetization factor and mutual coupling between different pieces of the sample. The asymmetry (panel (b)) decreases upon cooling since LSCO x=0.125 is known to have a magnetic phase concomitant with the superconducting one [3][4][5]. The muon spin relaxation (panel (c)) has a peak at the critical temperature, which is also unusual.
The presence of magnetism could be detrimental to our analysis if it depends on depth.
To verify that this is not the case, we perform zero field (ZF) measurements for different implantation energies at T = 5 K well below T c and for T = 30 K above T c . The results are presented in Supplementary Fig. 10. Fast relaxation and reduction of the asymmetry are observed at low temperature due to local random fields originating from the magnetic stripes in the sample. Nevertheless, there is no change in the magnetic relaxation with implantation depth.
The more sophisticated analysis method is presented in the main text. For each temperature, we fit all data sets with energy larger than 5 keV due to the presence of a dead layer, using Eq. 5. In the fit A 0 is a free parameter, and λ, u and B 0 are shared. A 0 is free because the number of muons actually penetrating the sample varies with energy. u represents relaxation processes that are implantation depth independent such as magnetism or field variations perpendicular to x. These are taken into account as some Lorentzian prob-