Signature of magnetic-dependent gapless odd frequency states at superconductor/ferromagnet interfaces

The theory of superconductivity developed by Bardeen, Cooper and Schrieffer (BCS) explains the stabilization of electron pairs into a spin-singlet, even frequency, state by the formation of an energy gap within which the density of states is zero. At a superconductor interface with an inhomogeneous ferromagnet, a gapless odd frequency superconducting state is predicted, in which the Cooper pairs are in a spin-triplet state. Although indirect evidence for such a state has been obtained, the gap structure and pairing symmetry have not so far been determined. Here we report scanning tunnelling spectroscopy of Nb superconducting films proximity coupled to epitaxial Ho. These measurements reveal pronounced changes to the Nb subgap superconducting density of states on driving the Ho through a metamagnetic transition from a helical antiferromagnetic to a homogeneous ferromagnetic state for which a BCS-like gap is recovered. The results prove odd frequency spin-triplet superconductivity at superconductor/inhomogeneous magnet interfaces.


Supplementary Figure 3. Resistance vs temperature (R-T) plots for a Au(3 nm)/Nb(20 nm)/ Ho(9.5 nm)/Nb(6.5 nm) film on sapphire and a Au(3 nm)/Nb (20 nm)/Nb (6.5 nm) on sapphire.
Electrical transport measurements were performed using a custom-built dipstick probe in a liquid helium dewar in a four-point current-bias setup. From the temperature dependence of a film's resistance, the superconducting transition T c was defined as the temperature at 50% of the transition. Care was taken to ensure that the applied current (0.1 mA) had no effect on T c . Critical temperatures of 6.5 K with Ho and 8.5 K without Ho were extracted meaning the proximity effect of Ho reduces T c by 2 K. The R-T plots were measured in increasing temperature. Evolution of the density of state spectrum with zero peak as a function of increasing temperature and (inset) zero-peak amplitude versus temperature. Figure 9. Effect of an out-of-plane magnetic field on a zero peak acquired on Au(3 nm)/Nb(20 nm)/Ho(9.5 nm)/Nb(6.5 nm) film on sapphire. The zero peak disappears below 500 mT and does not split into two peaks as expected if the zero peak is related to Kondo effects.

Supplementary
To ensure local reproducibility of the spectral features observed in our experiments and to determine any variation of conductivity on sample areas with topographic defects used as a reference, we have determined the conductance maps corresponding to the 16x16 DOS spectra matrix acquired on the 96 nm x 96 nm area of Au/Nb/Ho/Nb sample as given in Fig.  3 of the main text.
The conductance maps have been measured at voltages of 0 mV, -0.35 mV, -0.5 mV and -1.5mV. These specific voltages are necessary to identify the type of spectra (among the four possible ones shown in Figs. 3c to 3f of the main text) corresponding to each of the 16x16 matrix data points for the following reasons:  The conductance (G) at 0 mV (denoted G 0 ) can be used to differentiate between a BCS or double-peak spectra where G 0 ~ 0, a single-peak or noisy spectra where G 0 >>0);  The conductance value at -0.35 mV (denoted G -0.35 ) corresponds to the maximum conductance of a double-peak subgap feature and it can therefore be used to identify double-peak spectra;  The conductance value at -0.5 mV (denoted G -0.5 ) for all the DOS spectra that can be related to a double-peak subgap feature (G -0.35 >>0) can help distinguish between those representative of a rounded U-shaped-like BCS gap (G -0.5 > G -0.35 ) and those related to a double-peak (G -0.5 < G -0.35 );  The conductance at -1.5mV (G -1.5 mV ) is normally much larger than 1 for good spectra (double-peak, single-peak, BCS), but close to 1 for noisy spectra.
Supplementary Figure 6 helps visualize the above by showing the positions of G 0mV , G -0.35mV , G -0.5mV , G -1.5mV in the four types of DOS spectra measured in our experiment.
The normalized conductance maps measured at 0 mV, -0.35 mV, -0.5 mV and -1.5 mV are reported in Supplementary Figure 5. The colour scale is the same for all maps, e.g. light green corresponds to dI/dV = 1 in all maps. Unlike for the spectral map reported in Fig.  3b of the main text, which gives a concise overview on the spatial distribution of the different kinds of spectra along with its correlation to the local sample topography, a comparison between corresponding pixels on different conductance maps is here needed to collect the same kind of information.
A few distinct areas, however, are easy to identify in these. For instance, the area around X = -20 nm and Y = -20 nm is clearly noisy, since the conductance is equal to one independently on the voltage, as expected for a noisy spectrum (see Supplementary Figure  6a). The area around X= 20 nm and Y= 10 nm instead is dominated by BCS-like spectra, since the conductance remains very low from 0 mV to -0.5 mV (Supplementary Figures 5a to  5c) and then it becomes significantly larger than 1 at -1.5 mV. Similarly, the maps show that the area around X = 10 nm and Y =-40 nm is characterized by a conductance close to zero at 0 mV, increasing around -0.35 mV and then decreasing again at -0.5 mVwhich is clearly indicative of a region where spectra with double-side peaks have been recorded. All these observations are clearly in accordance with the spectral map reported in Fig. 3b of the manuscript and hence confirm its accurateness in describing the spectral distribution on the investigated sample area.
To model the experimental data we have performed calculations using the quasiclassical theory of superconductivity in the regime of diffusive transport by using the Usadel equation 1 . The equilibrium properties of the Nb/Ho system under consideration are expected to be well described by the Green's functions g S and g F in the superconductor (S) and ferromagnetic (F) regions, respectively. In order to uncover the odd-frequency correlations in Nb, it is necessary to solve the Usadel equation in both the S and F regions in addition to the self-consistent gap equation describing the superconducting order parameter Δ in S. This procedure is more complex than the usual approach in the literature where the bulk solution for g S is used, allowing for a simplification of the problem as only the Usadel equation in the F region has to be solved.
The two equations to solve for the Green's functions read: where ∆=∆(x) is the superconducting order parameter. The notation […] denotes a commutator. These equations are to be solved with the following boundary conditions at the S/F interface: These boundary conditions include the presence of a spin-active interface 2 captured by the parameters τ F(S),φ for a weakly polarized ferromagnet where the matrix α depends on the orientation of the interface moment relative to the bulk F. The parameter τ F(S) is a measure of the interface transparency defined as R B /R S(F) with R B being the resistance of the barrier region and R S(F) the normal-state resistance of the S(F) regions. At the outer edges of the S and F regions, the boundary conditions read ∂ x g = 0. Finally, one also has to take into account the self-consistency equation for the order parameter in the S region: where g K = (g S -g A )tanh(βε /2) and g A = -(ρ 3 g S ρ 3 ) † with β = 1/(k B T), T being the temperature and k B the Boltzmann constant. The coefficients N and V denote the DOS at the Fermi level in the normal-state of the superconductor and the net attractive interaction between the electrons causing Cooper pair formation. The matrices ρ 1 , ρ 2 , and τ 3 are 4x4 matrices which are given in Ref. 3.
The above system of equations is solved by an iterative schedule as follows. One first solves the Usadel equation for g F using an initial guess for the Green's function g S in the boundary condition for g F . The resulting solution for g F is then used to solve the Usadel equation for g S , which in turn is used as input for the self-consistent equation for . The procedure then starts over using the latest solution for g S and is repeated until the change in value obtained for ∆(x) does not exceed a small threshold value ϵ. In this way, one may verify that the obtained solutions for g F and g S simultaneously solve the two Usadel equations and the self-consistency equation for the gap.
In our numerical implementation, we made use of the Ricatti-parametrization 4,5 for the Green's function matrix and added a small imaginary value iδ with δ ≪ 1 to the quasiparticle energies  in order to account for inelastic scattering and pair breaking processes in the bulk S.