SUPPLEMENTARY INFORMATION. Magnetic field induced dissipation free state in superconducting nanostructures

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1 SUPPLEMENTARY INFORMATION Magnetic field induced dissipation free state in superconducting nanostructures R. Córdoba, 1, 2 T. I. Baturina, 3, 4 A. Yu. Mironov, 3 J. Sesé, 1, 2 J. M. De Teresa, 5, 2 M. R. Ibarra, 1, 5, 2 D. A. Nasimov, 3 A. K. Gutakovskii, 3 A. V. Latyshev, 3 I. Guillamón, 6, 7 H. Suderow, 6 S. Vieira, 6 M. R. Baklanov, 8 J. J. Palacios, 9 and V. M. Vinokur 4 1 Instituto de Nanociencia de Aragón, Universidad de Zaragoza, Zaragoza, 50009, Spain 2 Departamento de Física de la Materia Condensada, Universidad de Zaragoza, Zaragoza, Spain 3 A.V. Rzhanov Institute of Semiconductor Physics SB RAS, 13 Lavrentjev Avenue, Novosibirsk, Russia 4 Materials Science Division, Argonne National Laboratory, Argonne, Illinois 60439, USA 5 Instituto de Ciencia de Materiales de Aragón, Universidad de Zaragoza-CSIC, Facultad de Ciencias, Zaragoza, 50009, Spain 6 Laboratorio de Bajas Temperaturas, Departamento de Física de la Materia Condensada Instituto de Ciencia de Materiales Nicolás Cabrera, Facultad de Ciencias Universidad Autónoma de Madrid, E Madrid, Spain 7 H.H. Wills Physics Laboratory, University of Bristol, Tyndall Avenue, Bristol, BS8 1TL, United Kingdom 8 IMEC Kapeldreef 75, B-3001 Leuven, Belgium 9 Departamento de Física de la Materia Condensada Universidad Autónoma de Madrid, E Madrid, Spain Supplementary Figure S 1: (a) Magnetoresistive oscillations in the TiN film-based superconducting wire network (SWN). The numbers of the R axis correspond to the 100 mk curve, all other curves are shifted upwards, without changing the scale, for clarity. The period of oscillations is 50 mt corresponding to the superconducting flux quantum Φ 0 = π /e per unit cell. (b) The I-V characteristic in the log-log scale demonstrating clearly the ohmic behaviour extending up to 10 na. The inset shows the differential resistance vs. the dc current plot in the zero magnetic field.

2 Supplementary Figure S 2: Study of the 5 nm thin atomic-layer-deposition-grown TiN film by High Resolution Transmission Electron Microscopy. (a) The plan-view bright field image of the film. The film is polycrystalline with the densely-packed crystallites and continuous, i.e. no pinholes present. (b) Transmission electron-diffraction pattern of the film. The rings are characteristic to polycrystalline structures and correspond to fully stoichoimetric composition of the TiN film, while the point reflexes making up a square lattice image the crystalline lattice of the Si-substrate. (c) The cross-section micrograph of the film revealing the atomically smooth interface between the substrate and the TiN film and the atomically smooth surface of the TiN film. 2

3 3 SUPPLEMENTARY DISCUSSION Motion of a single vortex across the strip in London approximation. At B < B V, the average distance between vortices crossing the strip a > w, and their interaction is effectively screened. This implies that the behaviour of R(B) is controlled by the activation energy barrier for a single vortex. The energy of the vortex traversing the strip can be well described within the London approximation, which, rigorously speaking, applies to the strips where w ξ d, provided one takes into account properly the energy of the vortex core. Then the energy of a vortex as a function of its position across the strip is [S36,S37,S38 ]: E(x) = Φ2 0d (4πλ) 2 ln [ 2w πξ d sin ( πx ) ] + β w Φ 0Bs x(w x), (1) 8πλ2 where x is the lateral position of the vortex, and β = 0.38 accounts for the contribution of the vortex core found from the Ginzburg-Landau calculations [S36 ]. Plugging in the parameters of our SW we estimate the height of the barrier at very low field as E(w/2)/k B = 88.7 K at T = 3.6 K, and E(w/2)/k B = 117 K at T = 3.4 K. The second term of Eq. 1 yields the exponential growth of the resistance R exp(b/b ) with the field, where B = k B T [32πλ 2 /(Φ 0 w 2 ξ)]. Using the SW parameters we obtain B = T at T = 4 K in a fairly good agreement with the experimental results. Activation energy for a phase slip in a wire. At B > B V vortices form a chain along the median line of the strip. In a strongly disordered superconductor the vortex core can be viewed as a cylinder of a normal metal with the density of states ν N. Then one can describe a resistive stripe around the median line of the wire where vortices are confined as a normal phase with the effective density of the quasiparticle states ν(b) (B/B c3 )ν N, where the factor (B/B c3 ) is merely the fraction of the strip volume occupied by vortex cores; see the upper panel in the inset of Figure 4 of the publication, sketching the median effectively-metallic stripe shunted by the edge superconducting channels. At B = B c3 superconductivity vanishes and ν(b) = ν N. To estimate the energy of the phase slip, note that the number of states in the energy interval is ν(b) each carrying the energy, and, therefore, the density of the phase slip energy is 2ν(B) 2 2(B/B c3 )ν N 2. Further, the effective width of the SC channel is the magnetic length l B = ( /eb) 1/2 ξ d and the longitudinal size of the phase slip nucleus is ξ d. The phase slip energy, E PS, therefore, acquires the form E PS E C F(B), (2) where E C = 2ν N 2 ξd 2 s is the condensation energy specific to a given material, and F(B) is a universal function of the magnetic field. At B well below B c3, F(B) B/B c3. To determine the behaviour of F(B) near B c3, we note that in this region vortex cores well overlap and vortices strongly attract each other. Therefore to obtain the energy barrier, one can adopt a general theory of thermally activated motion of tightly coupled chain of particles [S39 ]and find F(B) (1 B/B c3 ) 5/4 at B B c3. While the exact form of scaling function F(B) should be found from numerical calculations, one can write down the interpolation F(B) (B/B c3 )(1 B/B c3 ) 5/4 as a fair description of the activation barrier in the range B max < B < B c3. Note that E C (T ) can be as large as several hundred T c giving rise to pretty complete suppressing the dissipation around B min. The decrease of the resistance after the maximum in Figure 2 of the publication and the subsequent increase at higher fields are thus the regions where the behaviour of F(B) is dominated by factors B/B c3 and (1 B/B c3 ) 5/4, respectively. To summarize, our equations 1 and 2 describe the N-shape magnetoresistance over the full range of the fields explored in the experiment. The equilibrium vortex pattern and the surface barrier in a wire. The equilibrium distribution of the superconducting order parameter Ψ in a wire is determined by minimizing the Ginzburg-Landau functional: [ G = dv Ψ ˆΠΨ + α Ψ 2 + β 2 Ψ 4 + ( b H) ] 2, (3) 8π with G = G s G n where G s and G n are the total Gibbs free energies for the superconductor and normal state, respectively, and ˆΠ = (1/2m )[ i (e /c) A] 2 is the kinetic energy operator. The magnetic field is aligned along the z-axis, the wire is aligned along the y direction, and the superconducting wire base surfaces are parallel to the xy plane. The effective (Pearl) penetration length Λ = λ 2 /d is greater than the width of the sample w (the x axis is taken across the wire), therefore the screening of the magnetic field is negligible. One then takes b( r) = B = Bẑ, where B is the applied field, and ẑ is the unit vector along the z direction. The lengths and energies are measured

4 4 T (K) ξ (nm) B V (T) B c3 (T) Supplementary Table S 1: Parameters of the wire. Superconducting coherence length ξ, and calculated critical fields (entry of vortices at B c1, and the upper field B c3) at different temperatures. in in the units of the magnetic length l B = c/(e B) and the lowest Landau level energy E c = ω c /2, respectively, where ω c = e B/(m c) is the cyclotron frequency. Presenting the order parameter as a linear combination of eigenfunctions φ k of the kinetic operator with eigenenergies ɛ k and using the Landau gauge for the vector potential A = Bxŷ, where ŷ is the unit vector along the y direction, one finds the following expression for the (normalized) eigenfunctions: φ k = 1 Ld e iky χ k (x). (4) Here L is the length of the wire and d is its thickness (L ), k is a set of the wavevectors in the y direction, and χ k (x) are the lowest Landau level eigenfunctions, i.e., eigenfunctions without nodes, numerically evaluated with the standard zero-current boundary condition across a superconductor-vacuum interface and the periodic boundary conditions in y direction. Upon expanding the order parameter over the basis of χ-functions the free-energy functional assumes the form G = C k 2 α k + β Ck 1 Ck 2 C k3 C k4 dxχ k1 χ k2 χ k3 χ k4 δ k3+k 4,k 1+k 2, (5) k k 1,k 2,k 3,k 4 where α k = α + ɛ k is the condensation energy corresponding to the k-th mode and β = β/(2l y L z le c ) = 2πe κ 2 /(LdlBm c) is a dimensionless parameter that determines G as a function of the physical dimensions of the sample. Then G V = (B/B c3 ) G represents the energy per vortex. Carrying out the minimization procedure (the details are given in Ref. [S40 ]), and taken the dimensions of the wire used in the experiment, one finds that at T < 3.8 K the wire contains just one vortex row of vortices and that at T > 3.8 K vortices do not appear in the wire at all. The difference k = k 2 k 1 between the wavevectors in the expansion determines the distance between the vortices in a chain, y = 2π k. The superconducting parameters and the calculated critical fields for various temperatures are presented in Table 1. The coherence lengths ξ were determined from the experimental data. The lower critical field, B V, is the lowest field at which vortices in the wire become thermodynamically stable. The upper critical field, B c3, is the field above which superconductivity in the wire ceases to exist. By varying the wavevectors while keeping k fixed, one can freely fix the vortex row position across the strip and calculate the free energy per vortex as a function of this position, G V (x). This is to be compared with the value of the thermodynamically stable state, G(0), where vortices are located at the wire median line [S40,S41,S42 ], as a function of the magnetic field and for different temperatures. We identify the barrier height for the vortex motion in the presence of a current from the largest difference. Note that this calculation neglects the reduction of the barrier due to the repulsion forces exerted by the chain on a vortex deviating from the median line, i.e. the determined barrier overestimates the true barrier for the vortex exit. The latter corresponds to the tongue-like saddle configuration which the vortex chain assumes in the course of the thermally activated escape from the wire under the applied current, but derivation of the saddle configuration and its energy goes beyond the scope of this work. Yet, the obtained results capture the non-monotonic dependence of the escape barrier upon the magnetic field, and are expected to differ from the true values not more than by the factor of two.

5 5 Supplementary References S36 G. Stejic, A. Gurevich, E. Kadyrov, D. Christen, R. Joint, & D. C. Larbalestier, Effect of geometry on the critical currents in thin films, Phys. Rev. B 49, (1994). S37 V. G. Kogan, Pearls vortex near the film edge, Phys. Rev. B 49, (1994). S38 G. M. Maksimova, Mixed state and critical current in narrow semiconducting films, Phys. Solid State, 40, 1610 (1998). S39 M. Büttiker, & R. Landauer, Nucleation theory of overdamped soliton motion, Phys. Rev., A 23, (1980). S40 J. J. Palacios, Phys. Rev. B 57, Vortex lattices in strong type-ii superconducting two-dimensional strips, (1998). S41 P. Sánchez-Lotero and J. J. Palacios, Critical fields for vortex expulsion from narrow superconducting strips, Phys. Rev. B 75, (2007). S42 P. Sánchez-Lotero and J. J. Palacios, Solutions of the Ginzburg-Landau functional with a current constraint, Physica C 404, (2004).