Quantum size effect on the paramagnetic critical field in Pb nanofilms

The quantum size effect on the in-plane paramagnetic critical field in Pb nanofilms is investigated with the use of the spin-generalized Bogolubov-de Gennes equations. It is shown that the critical field oscillates as a function of the nanofilm thickness with the period of $\sim2$ ML (even-odd oscillations) modulated by the beating effect. This phenomena is studied in terms of the quantization of the electron energy spectra caused by the confinement of the electron motion in the direction perpendicular to the sample. The calculated values of critical fields for different nanofilm thicknesses are analyzed in the context of Clogston-Chandrasekhar limit. The influence of the thermal effect on the magnetic field induced superconductor to normal metal transition is also discussed. Furthermore, the thickness-dependence of the electron-phonon coupling and its influence on the value of the critical magnetic field are studied.Comment: 9 pages, 9 figure

Fulde-Ferrell state in superconducting core/shell nanowires: role of the orbital effect

The orbital effect on the Fulde-Ferrell (FF) phase is investigated in superconducting core/shell nanowires subjected to the axial magnetic field. The confinement in the radial direction results in the quantization of the electron motion with energies determined by the radial $j$ and orbital $m$ quantum numbers. In the external magnetic field the twofold degeneracy with respect to the orbital magnetic quantum number $m$ is lifted which leads to the Fermi wave vector mismatch between the paired electrons $(k, j,m,\uparrow) \leftrightarrow (-k, j,-m,\downarrow)$. This mismatch is transfered to the nonzero total momentum of the Cooper pairs which results in the formation of FF phase occurring sequentially with increasing magnetic field. By changing the nanowire radius $R$ and the superconducting shell thickness $d$, we discuss the role of the orbital effect in the FF phase formation in both the nanowire-like ($R/d \ll 1$) and nanofilm-like ($R/d \gg 1$) regime. We have found that the irregular pattern of the FF phase, which appears for the case of the nanowire-like regime, evolves towards the regular distribution, in which the FF phase stability regions appear periodically between the BCS state, for the nanofilm-like geometry. The crossover between these two different phase diagrams is explained as resulting from the orbital effect and the multigap character of superconductivity in core/shell nanowires.Comment: 10 pages, 7 figure

Influence of the electron density on the thickness-dependent energy gap oscillations in superconducting metallic nanofilms

The thickness-dependent energy gap oscillations in the metallic nanofilms are investigated by the use of the self-consistent numerical solutions of the Bogoliubov-de Gennes equations. It is shown, that the oscillations are induced by the quasi-particle energy quantization triggered by the confinement of electrons in the direction perpendicular to the sample. We have analyzed, how the changes in the electron density of states ($n_e$) and the electron-phonon coupling constant ($g$) influence the amplitude of the considered oscillations. It has been found, that the increase in $n_e$ and the decrease in $g$, can lead to a significant reduction of the oscillations amplitude. As a result, for the values of the mentioned parameters corresponding to some of the realistic situations the thickness-dependent superconducting gap oscillations can be almost completely suppressed

Orbital effect on the in-plane critical field in free-standing superconducting nanofilms

The superconductor to normal metal phase transition induced by the in-plane magnetic field is studied in free-standing Pb(111) nanofilms. In the considered structures the energy quantization induced by the confinement leads to the thickness-dependent oscillations of the critical field (the so-called 'shape resonances'). In this paper we examine the influence of the orbital effect on the in-plane critical magnetic field in nanofilms. We demonstrate that the orbital term suppresses the critical field and reduces the amplitude of the thickness-dependent critical field oscillations. Moreover, due to the orbital effect, the slope $H_{c,||}-T_c$ at $T_c(0)$ becomes finite and decreases with increasing film thickness in agreement with recent experiments. The temperature $t^*$ at which the superconductor to normal metal phase transition becomes of the first order is also analyzed.Comment: 8 pages, 7 figure

Renormalization of the Majorana bound state decay length in a perpendicular magnetic field

Orbital effects of a magnetic field in a proximitized semiconductor nanowire are studied in the context of the spatial extent of Majorana bound states. We develop analytical model that explains the impact of concurring effects of paramagnetic coupling of the nanowire bands via the kinetic energy operator and spin-orbit interaction on the Majorana modes. We find, that the perpendicular field, so far considered as to be detrimental to the Majorana fermion formation, is in fact helpful in establishing the topological zero-energy-modes in a finite system due to significant decrease in the Majorana decay length

Probing Andreev reflection reach in semiconductor-superconductor hybrids by Aharonov-Bohm effect

Recent development in fabrication of hybrid nanostructures allows for creation of quantum interferometers that combine semiconductor and superconductor materials. We show that in those nanostructures the joint phenomena of Aharonov-Bohm effect and Andreev reflections can be used to determine the length on which the electron is retro-reflected as a hole. We propose to exploit this feature for probing of the quasiparticle coherence length in semiconductor-superconductor hybrids by a magnetoconductance measurement

Nonseparably connected complete metric spaces

A topological space is nonseparably connected if it is connected but all of its connected separable subspaces are singletons. We show that each connected first countable space is the image of a nonseparably connected complete metric space under a continuous monotone hereditarily quotient map.Comment: This manuscript is not intended for publication because we have a more advanced version alread

Intersubband pairing induced Fulde-Ferrell phase in metallic nanofilms

We consider a free-standing metallic nanofilm with a predominant intersubband paring which emerges as a result of the confinement in the growth direction. We show that the Fermi wave vector mismatch between the subbands, detrimental to the intersubband pairing, can be compensated by the non-zero center of mass momentum of the Cooper pairs. This leads to the spontaneous appearance of the intersubband Fulde-Ferrell (IFF) state, even in the absence of an external magnetic field. Our study of the intrasubband pairing channel on the stability of the IFF phase shows that the former strongly competes with the intersubband pairing, which prohibits the coexistence of the two superconducting phases. Interestingly, upon application of the magnetic field we find a transition to an exotic mixed spin-singlet subband-triplet and spin-triplet subband-singlet paired state. Finally, we discuss the possibility of existence of the IFF pairing in novel superconducting materials.Comment: 8 pages, 5 figure

Connected, not separably connected complete metric spaces

In a separably connected space any two points are contained in a separable connected subset. We show a mechanism that takes a connected bounded metric space and produces a complete connected metric space whose separablewise components form a quotient space isometric to the original space. We repeatedly apply this mechanism to construct, as an inverse limit, a complete connected metric space whose each separable subset is zero-dimensional.Comment: 10 page

Tunneling conductance in half-metal/conical magnet/superconductor junctions in the adiabatic and non-adiabatic regime: self-consistent calculations

The tunneling conductance in the half-metal/conical magnet/superconductor (HM/CM/SC) is investigated by the use of the combined Blonder-Tinkham-Klapwijk (BTK) formalism and the Bogoliubov-de Gennes (BdG) equations. We show that the conductance calculated self-consistently differs significantly from the one calculated in the non-self-consistent framework. The use of the self-consistent procedure ensures that the charge conservation is satisfied. Due to the spin band separation in the HM, the conductance in the subgap region is mainly determined by the anomalous Andreev reflection the probability of which strongly depends on the spin transmission in the CM layer. We show that the spin of electron injected from the HM can be transmitted through the CM to the SC adiabatically or non-adiabatically depending on the period of the exchange field modulation. We find that the conductance in the subgap region oscillates as a function of the CM layer thickness wherein the oscillations transform from irregular, in the non-adiabatic regime, to regular in the adiabatic case. In the non-adiabatic regime the decrease of the exchange field amplitude in the CM leads to the emergence of the conductance peak for one particular CM thickness in agreement with experiment [J.W.A Robinson, J. D. S Witt and M. G. Blamire, Science 329, 5987]. For both transport regimes the conductance is analyzed over a broad range of parameters determining the spiral magnetization in the CM.Comment: 12 pages, 16 figure