Exploring disorder correlations in superconducting systems: spectroscopic insights and matrix element effects

• Vyacheslav D. Neverov,
• Alexander E. Lukyanov,
• Andrey V. Krasavin,
• Alexei Vagov,
• Boris G. Lvov and
• Mihail D. Croitoru

Beilstein J. Nanotechnol. 2024, 15, 199–206, doi:10.3762/bjnano.15.19

• demonstrated that power-law correlations alter both the spatial distribution and the absolute value of the superconducting order parameter. Notably, an increase in the degree of correlation within the disorder potential is shown to augment superconductivity, aligning closely with experimental findings

• superconducting order parameter Δi and the Hartree potential Ui are determined through the self-consistency equations [6]: where the angular brackets ⟨…⟩ denote quantum mechanical averages, and the sum goes over the eigenfunctions of the BdG equations, labeled by index n. The Hartree self-consistency condition

• take N = 50, which our calculations show is sufficiently large to mitigate finite-size effects. BdG equations (Equation 2) are solved together with the self-consistency conditions (Equation 3) in the usual way until the order parameter and the Hartree potential reach a predefined accuracy threshold [63

Josephson dynamics and Shapiro steps at high transmissions: current bias regime

• Artem V. Galaktionov and
• Andrei D. Zaikin

Beilstein J. Nanotechnol. 2024, 15, 51–56, doi:10.3762/bjnano.15.5

• ) electrodes is kept much shorter than the superconducting coherence length, that is, d ≪ ξ0 ∼ vF/Δ. Here, Δ is the absolute value of the order parameter in superconducting electrodes and vF denotes the Fermi velocity. For simplicity here and below, we set the Planck and Boltzmann constants equal to unity (ℏ

Spin dynamics in superconductor/ferromagnetic insulator hybrid structures with precessing magnetization

• Yaroslav V. Turkin and
• Nataliya Pugach

Beilstein J. Nanotechnol. 2023, 14, 233–239, doi:10.3762/bjnano.14.22

• formalism of two-time quasiclassical Green’s functions in Nambu–spin-Keldysh space [28]. We expand the Green’s function assuming a weak proximity effect [28] with the ferromagnetic insulator: . To handle the expansion of the order parameter correctly, we should cancel the odd orders of the perturbation

• , because the triplet Green’s function components do not contribute to the order parameter. Only even orders of the perturbation series determine its correction. Thus, the superconducting order parameter in the linear regime, has only a zero-order term in expansion. The resulting dynamics of the

• superconducting condensate in the weak proximity effect regime can be described via the nonstationary Usadel equation [18][19][30]: where is the stationary BCS superconducting order parameter matrix [28], D is the diffusion constant, is the auxiliary matrix in Nambu–spin-Keldysh space, is the time convolution

Cooper pair splitting controlled by a temperature gradient

• Dmitry S. Golubev and
• Andrei D. Zaikin

Beilstein J. Nanotechnol. 2023, 14, 61–67, doi:10.3762/bjnano.14.7

• projection α at a point x, m is the electron mass, and μ is the chemical potential, is the Hamiltonian of a superconducting electrode with the order parameter Δ and the terms account for electron transfer through the junctions between the superconductor and the normal leads. In Equation 4, the surface

Upper critical magnetic field in NbRe and NbReN micrometric strips

• Zahra Makhdoumi Kakhaki,
• Antonio Leo,
• Federico Chianese,
• Loredana Parlato,
• Giovanni Piero Pepe,
• Angela Nigro,
• Carla Cirillo and
• Carmine Attanasio

Beilstein J. Nanotechnol. 2023, 14, 45–51, doi:10.3762/bjnano.14.5

• line with other studies on Hc2⟂(T) made on non-centrosymmetric materials in bulk forms [37][39][41], may suggest the presence of a triplet component of the order parameter. This result was even more evident in the case of polycrystalline NbRe samples, for which the experimental points in the H–T phase

• in the atomic cell may break the non-centrosymmetricity of the system, thus depressing the spin-triplet component of the order parameter. For this reason, Hc2⟂ becomes paramagnetically limited [24][35]. In order to confirm these results, we are currently working on different experiments that may give

• more direct access to the order parameter in these systems. Regarding the aforementioned purpose, while in the case of NbRe it is now even more evident that films with larger crystallites are mandatory [7][8][12], detailed analyses of the NbReN crystal structure are still lacking and need to be

Density of states in the presence of spin-dependent scattering in SF bilayers: a numerical and analytical approach

• Tairzhan Karabassov,
• Valeriia D. Pashkovskaia,
• Nikita A. Parkhomenko,
• Anastasia V. Guravova,
• Elena A. Kazakova,
• Boris G. Lvov,
• Alexander A. Golubov and
• Andrey S. Vasenko

Beilstein J. Nanotechnol. 2022, 13, 1418–1431, doi:10.3762/bjnano.13.117

• Usadel equation has the following form [93]: Here, Ds is the diffusion coefficient in the superconductor, and Δ(x) is the superconducting order parameter (pair potential). From the Usadel equations, it can be shown that there is a symmetry relation between θ↑ and θ↓: θ↑(E) = (−E), where E is the energy

• (ωn → − iE) and * is the complex conjugation. Equation 2 and Equation 3 should be supplemented with the self-consistency equation for the coordinate dependence of superconducting order parameter Δ, The resulting system must be complemented by the boundary conditions at the outer boundary of a

• the strength of superconductivity suppression in the S layer by the ferromagnet F (inverse proximity effect). For instance, when γ ≫ 1, the inverse proximity effect is very strong, and the order parameter is heavily suppressed near the SF interface compared to its bulk value. On the contrary, when γ

Experimental and theoretical study of field-dependent spin splitting at ferromagnetic insulator–superconductor interfaces

• Peter Machon,
• Michael J. Wolf,
• Detlef Beckmann and
• Wolfgang Belzig

Beilstein J. Nanotechnol. 2022, 13, 682–688, doi:10.3762/bjnano.13.60

• values of ε’ = [100,0.1]. Density of states in a superconductor in proximity to a ferromagnetic insulator indicated by the spin mixing angle δφ, with (from (a) to (d)) ε’ = 100,10,1,0.1, while the superconducting order parameter is evaluated self-consistently (T ≪ Tc). False-color scanning electron

Tunable superconducting neurons for networks based on radial basis functions

• Andrey E. Schegolev,
• Nikolay V. Klenov,
• Sergey V. Bakurskiy,
• Igor I. Soloviev,
• Mikhail Yu. Kupriyanov,
• Maxim V. Tereshonok and
• Anatoli S. Sidorenko

Beilstein J. Nanotechnol. 2022, 13, 444–454, doi:10.3762/bjnano.13.37

• proximity effect. The typical spin valve [55][56][57] is a hybrid structure containing at least a pair of ferromagnetic (FM) layers with different coercive forces. Variations in the relative orientation of their magnetizations change the spatial distribution of the superconducting order parameter. In the

• case of parallel magnetization of the FM layers the Cooper pairs are effectively depairing inside them (closed spin valve). For the antiparallel orientation, the effective exchange energy of the magnetic layers is averaged and suppression of the superconducting order parameter is weaker (open spin

• layers supports the superconducting order parameter and increases the efficiency of the spin valve effect [58]. Here, we propose a development of this approach, allowing one to significantly increase the effective variations in the kinetic inductance. We study proximity effect and electronic transport in

Plasma modes in capacitively coupled superconducting nanowires

• Alex Latyshev,
• Andrew G. Semenov and
• Andrei D. Zaikin

Beilstein J. Nanotechnol. 2022, 13, 292–297, doi:10.3762/bjnano.13.24

• prominent role of fluctuation effects in a reduced dimension [1][2][3]. Such fluctuations cause a reduction of the superconducting critical temperature [4] and yield a negative correction to the mean field value of the order parameter Δ0. In particular, at T→0 for the absolute value of the order parameter

• fluctuation correction to the mean value of the superconducting order parameter (Equation 1) remains weak and in the majority of cases can be neglected. Is the condition Rξ/Rq ≪ 1 sufficient to disregard fluctuation effects in superconducting nanowires? The answer to this question is obviously negative since

• even in this limit fluctuations of the phase φ(x,t) of the order parameter Δ = |Δ|exp(iφ) survive being essentially decoupled from those of the absolute value |Δ|. Such phase fluctuations are intimately related to sound-like plasma modes [6][7] (the so-called Mooij–Schön modes), which can propagate

In situ transport characterization of magnetic states in Nb/Co superconductor/ferromagnet heterostructures

• Olena M. Kapran,
• Roman Morari,
• Taras Golod,
• Evgenii A. Borodianskyi,
• Vladimir Boian,
• Andrei Prepelita,
• Nikolay Klenov,
• Anatoli S. Sidorenko and
• Vladimir M. Krasnov

Beilstein J. Nanotechnol. 2021, 12, 913–923, doi:10.3762/bjnano.12.68

• unconventional odd-frequency spin-triplet order parameter should appear. The non-hysteretic nature of this state allows for reversible tuning of the magnetic orientation. Thus, we identify the range of parameters and the procedure for in situ control of devices based on S/F heterostructures. Keywords: cryogenic

• the possible generation of the odd-frequency spin-triplet order parameter [1][2][3]. In recent years, this exotic state has been extensively studied both theoretically [4][5][6][7][8][9][10][11][12][13][14][15][16][17] and experimentally [18][19][20][21][22][23][24][25][26][27][28][29][30][31][32][33

• application perspective, the most important is the noncollinear scissor state, in which the unconventional odd-frequency spin-triplet order parameter should appear. The non-hysteretic nature of this state allows for reversible tuning of the magnetic configuration. Thus, we identify the range of parameters and

Electromigration-induced formation of percolating adsorbate islands during condensation from the gaseous phase: a computational study

• Alina V. Dvornichenko,
• Vasyl O. Kharchenko and
• Dmitrii O. Kharchenko

Beilstein J. Nanotechnol. 2021, 12, 694–703, doi:10.3762/bjnano.12.55

• panel) and the dispersion ⟨(δx)2⟩ (bottom panel). The dispersion ⟨(δx)2⟩ = ⟨x2⟩ − ⟨x⟩2 is an order parameter for pattern formation. If ⟨(δx)2⟩ ≃ 0 then the field x(r) is homogeneously distributed and no patterns are possible. The growing dynamics ⟨(δx)2⟩(t) indicates ordering of the field x(r) with

Superconductor–insulator transition in capacitively coupled superconducting nanowires

• Alex Latyshev,
• Andrew G. Semenov and
• Andrei D. Zaikin

Beilstein J. Nanotechnol. 2020, 11, 1402–1408, doi:10.3762/bjnano.11.124

• ]. Many interesting properties of such nanowires are attributed to the effect of quantum phase slips (QPSs) which correspond to fluctuation-induced local temporal suppression of the superconducting order parameter inside the wire accompanied by the phase slippage process and quantum fluctuations of the

• and below Rq = 2π/e2 ≃ 25.8 kΩ is the quantum resistance unit and Rjξ is the normal state resistance of the corresponding wire segment), Δ is the superconducting order parameter and a ≈ 1 is a numerical prefactor. We also note that the Hamiltonian (Equation 5) describes tunneling of the magnetic flux

Controlling the proximity effect in a Co/Nb multilayer: the properties of electronic transport

• Sergey Bakurskiy,
• Mikhail Kupriyanov,
• Nikolay V. Klenov,
• Igor Soloviev,
• Andrey Schegolev,
• Roman Morari,
• Yury Khaydukov and
• Anatoli S. Sidorenko

Beilstein J. Nanotechnol. 2020, 11, 1336–1345, doi:10.3762/bjnano.11.118

• the conditions at which switching from the parallel to the antiparallel alignment of the neighboring F-layers leads to a significant change of the superconducting order parameter in superconductive thin films. We experimentally study the transport properties of a lithographically patterned Nb/Co

• ferromagnetic (F) layers separated by thin superconducting layers, in which the superconducting order parameter is maintained due to the proximity to a thick superconducting bank (S-bank). Switching from the antiparallel (AP) to the parallel (P) alignment of neighboring F1 and F2 layers leads to a significant

• which the thick S-bank acts as the source of induced superconductivity, is the simplest model of the 3D structure. Let us consider the applications that are possible due to the control over the order parameter in thin superconductor layers (s-layers) in such a structure. The simplest cell for the

Nonadiabatic superconductivity in a Li-intercalated hexagonal boron nitride bilayer

• Kamila A. Szewczyk,
• Izabela A. Domagalska,
• Artur P. Durajski and
• Radosław Szczęśniak

Beilstein J. Nanotechnol. 2020, 11, 1178–1189, doi:10.3762/bjnano.11.102

• allow for the self-consistent determination of the superconducting order parameter (Δn = Δ(iωn) and the wave function renormalization factor (Zn = Z(iωn), with an accuracy of the second order relative to the electron–phonon coupling function (g). The symbol ωn = πkBT(2n + 1) defines the fermionic

• –Eliashberg equations. The order parameter is given by the formula Δn = φn/Zn. The symbol λn,m represents the pairing kernel for the electron–phonon interactions: The Coulomb pseudopotential function is: , where θ(x) is the Heaviside function, and ωc represents the cut-off frequency (ωc = 3ωD = 496.7 meV

• ). Freericks’ equations allow one to determine the values of the order parameter and the wave function renormalization factor in a self-consistent manner, which is undoubtedly their great advantage. These are isotropic equations, which means that the self-consistent procedure does not apply to the electron

A Josephson junction based on a highly disordered superconductor/low-resistivity normal metal bilayer

• Pavel M. Marychev and
• Denis Yu. Vodolazov

Beilstein J. Nanotechnol. 2020, 11, 858–865, doi:10.3762/bjnano.11.71

• . Here ℏωn = πkBT(2n + 1) are the Matsubara frequencies (n is an integer number), q = ∇ϕ = (qx, qz) is the quantity that is proportional to the supervelocity vs, and ϕ is the phase of the superconducting order parameter. Δ is the magnitude of the order parameter, which should satisfy the self-consistency

• found from the self-consistings solution of Equation 1–Equation 3 and Equation 5. In numerical calculations we use dimensionless units. The magnitude of the order parameter is normalized by kBTc0 = Δ(0)/1.76, lengths are in units of ≈ 1.33ξ(0), where is the superconducting coherence length at T = 0

• . Interestingly, in contrast to common junctions, Ic increases in the SN-S-SN system. This can be explained by a lower value of the superconducting order parameter in SN banks in comparison with Δ in the S constriction at Is = 0. With increasing a the superconducting order parameter in the constriction increases

Anomalous current–voltage characteristics of SFIFS Josephson junctions with weak ferromagnetic interlayers

• Tairzhan Karabassov,
• Anastasia V. Guravova,
• Aleksei Yu. Kuzin,
• Elena A. Kazakova,
• Shiro Kawabata,
• Boris G. Lvov and
• Andrey S. Vasenko

Beilstein J. Nanotechnol. 2020, 11, 252–262, doi:10.3762/bjnano.11.19

• -known bulk BCS form. We notice that the density of states at x = ±∞ is given by standard BCS equation, where Θ(x) is the Heaviside step function. Finally the self-consistency equation for the superconducting order parameter takes the form, Equations Equation 3–Equation 8 and Equation 10 represent a

Rational design of block copolymer self-assemblies in photodynamic therapy

• Maxime Demazeau,
• Laure Gibot,
• Anne-Françoise Mingotaud,
• Patricia Vicendo,
• Clément Roux and
• Barbara Lonetti

Beilstein J. Nanotechnol. 2020, 11, 180–212, doi:10.3762/bjnano.11.15

Transport signatures of an Andreev molecule in a quantum dot–superconductor–quantum dot setup

• Zoltán Scherübl,
• András Pályi and
• Szabolcs Csonka

Beilstein J. Nanotechnol. 2019, 10, 363–378, doi:10.3762/bjnano.10.36

• ) Hamiltonian where εSk is the dispersion of conduction electrons in the SC, cSkσ () annihilates (creates) an electron in the SC with momentum k and spin σ, and Δ is the superconducting order parameter. Only one SC lead is present in our setup, hence Δ is chosen to be real. HSC can be diagonalized by a

Threshold voltage decrease in a thermotropic nematic liquid crystal doped with graphene oxide flakes

• Mateusz Mrukiewicz,
• Krystian Kowiorski,
• Paweł Perkowski,
• Rafał Mazur and
• Małgorzata Djas

Beilstein J. Nanotechnol. 2019, 10, 71–78, doi:10.3762/bjnano.10.7

• nanoparticles of MgO and SiO2 [6]. Here, the effect was caused by the reduction of the order parameter S. The concentration-dependent enhancement of the electro-optic response was observed for Ti and TiO2 nanoparticles dispersed in NLC [7][8]. The significant effect of ferroelectric Sn2P2S6 on the threshold

Pinning of a ferroelectric Bloch wall at a paraelectric layer

• Vilgelmina Stepkova and
• Jiří Hlinka

Beilstein J. Nanotechnol. 2018, 9, 2356–2360, doi:10.3762/bjnano.9.220

• paraelectric layer. Thus, the layer acts as a bottleneck for the helicity order parameter of the wall. Conclusion In summary, since the Bloch character is strongly suppressed when the domain wall is right at the SrTiO3 layer, the layer can facilitate selection of the sign of the Pt component, and therefore

Surface energy of nanoparticles – influence of particle size and structure

• Dieter Vollath,
• Franz Dieter Fischer and
• David Holec

Beilstein J. Nanotechnol. 2018, 9, 2265–2276, doi:10.3762/bjnano.9.211

• Johnson [35] do not show such a size limit. In contrast, the phenomenon of premelting at the surface enhances with decreasing particle size. Figure 6 displays Landau’s order parameter M [36] for different particle diameters as function of the radial distance from the particle center, denoted as radius [35

• ]. This order parameter obtains the value 1 for perfectly crystallized and 0 for liquid material. Certainly, such a surface layer with reduced crystalline order has a surface energy, which is closer to that of a liquid rather than a crystalline solid. Also this result supports arguments leading to the

• point. Molecular dynamics calculations using gold as an example resulted in a similar behavior. Qiao et al. [47] used, similarly as Chang and Johnson [35], an order parameter defined as the “translational order parameter” [48], which is 1 for perfectly crystallized materials and 0 for liquids, as a

Josephson effect in junctions of conventional and topological superconductors

• Alex Zazunov,
• Albert Iks,
• Miguel Alvarado,
• Alfredo Levy Yeyati and
• Reinhold Egger

Beilstein J. Nanotechnol. 2018, 9, 1659–1676, doi:10.3762/bjnano.9.158

• geometry: S denotes a conventional s-wave BCS superconductor with order parameter , and TS represents a topologically nontrivial superconducting wire with MBSs (shown as stars) and proximity-induced order parameter . The interface contains a quantum dot (QD) corresponding to an Anderson impurity, connected

• to the S/TS leads by tunnel amplitudes λS/TS (light red). The QD is also exposed to a local Zeeman field B. b) S–TS–S geometry: Two conventional superconductors (S1 and S2) with the same gap Δ and a TS wire with proximity gap Δp form a trijunction. The order parameter phase of S1 (S2), 1 = /2 (2

Interplay between pairing and correlations in spin-polarized bound states

• Szczepan Głodzik,
• Aksel Kobiałka,
• Anna Gorczyca-Goraj,
• Andrzej Ptok,
• Grzegorz Górski,
• Maciej M. Maśka and
• Tadeusz Domański

Beilstein J. Nanotechnol. 2018, 9, 1370–1380, doi:10.3762/bjnano.9.129

• where is the local superconducting order parameter and niσ = . The Hartree term can be incorporated into the local (spin-dependent) chemical potential μ → ≡ μ − . The second term in Equation 1 refers to the local impurity which affects the order parameter χi near the impurity site i = 0, inducing the

• -of-plane spin–orbit field, respectively, and satisfy . Solving numerically the BdG equations (Equation 6) we can determine the local order parameter χi and occupancy niσ where f(ω) = [1 + exp(ω/kBT)]−1. In what follows, we shall inspect the spin-resolved local density of states For its numerical

• the bound states for two representative values of the spin–orbit coupling λ upon varying J. The magnetic potential has substantial influence on the local order parameter χ0. In particular, at some critical value Jc this quantity discontinuously changes its magnitude and sign (see the upper panel in

Andreev spectrum and supercurrents in nanowire-based SNS junctions containing Majorana bound states

• Jorge Cayao,
• Annica M. Black-Schaffer,
• Elsa Prada and
• Ramón Aguado

Beilstein J. Nanotechnol. 2018, 9, 1339–1357, doi:10.3762/bjnano.9.127

• the order parameter is assumed to be zero as it is only relevant when investigating Andreev bound states in SNS junctions. In Figure 2 we present the low-energy spectrum for a superconducting nanowire as a function of the Zeeman field at a fixed wire length LS. Figure 2a shows the case of zero

Formation and development of nanometer-sized cybotactic clusters in bent-core nematic liquid crystalline compounds

• Yuri P. Panarin,
• Sithara P. Sreenilayam,
• Jagdish K. Vij,
• Anne Lehmann and
• Carsten Tschierske

Beilstein J. Nanotechnol. 2018, 9, 1288–1296, doi:10.3762/bjnano.9.121

• around the long axes needs to be broken by aligning one of the short axes or if the long axes need to be subjected to such a strong magnetic field. Can such a large field induce the orientational order parameter greater than its thermodynamic value? Nevertheless it is now clear that many of the