Diffusive Normal Metal Research Articles

Odd-frequency pairs and anomalous proximity effect in nematic and chiral states of superconducting topological insulators

We investigate emergent odd-frequency pairs and proximity effect in nematic and chiral states of superconducting topological insulators (STIs), such as ${M}_{x}{\mathrm{Bi}}_{2}{\mathrm{Se}}_{3}$ ($M=$ Cu, Sr, Nb). The interplay of pairing symmetry, the orbital degrees of freedom, and strong spin-orbit interaction generates a variety of odd-frequency pairs in the bulk and surface of STIs. The nematic and chiral states are the prototypes of topological superconductors with and without time-reversal symmetry, respectively. We find that the Fermi surface evolution from a closed spheroidal to an open cylindrical shape changes the pairing symmetry from the nematic to chiral state, which causes the evolution of the odd-frequency pairings and surface Andreev bound states (SABSs). In addition, spin polarization of odd-frequency pairs and SABSs stems from the nonunitary pairing in the chiral state. The evolution and spin polarization of odd-frequency pairs and SABSs can be captured by tunnel conductance spectroscopy. Furthermore, we study the anomalous proximity effect in various irreducible representations of STIs. The anomalous proximity effect was originally predicted in spin-triplet superconductor junctions without spin-orbit interaction. Odd-frequency spin-triplet $s$-wave pairs penetrate into diffusive normal (DN) metals and induce a pronounced zero-energy peak of the local density of states in the DN region. Here we demonstrate that contrary to the well-known results, the anomalous proximity effect in STIs is not immune to nonmagnetic impurities. The fragility is attributed to the fact that the proximitized odd-frequency even-parity pairs are admixtures of $s$-wave and non-$s$-wave pairs due to strong spin-orbit interaction inherent to the parent materials.

Generation of polarized spin-triplet Cooper pairings by magnetic barriers in superconducting junctions

We investigate the proximity effect in an s-wave superconductor/ferromagnetic metal with a Rashba spin-orbit coupling/diffusive normal metal junction and an s-wave superconductor/noncollinear magnetic metal/diffusive normal metal junction. We show the generation of equal spin-triplet pairings in the diffusive normal metal due to spin-flip scattering in the intermediate magnetic regions. The emergence of the spin-triplet odd-frequency Cooper pairings can generate a zero-energy peak in the quasiparticle density of states in the diffusive normal metal.

Probing the topological band structure of diffusive multiterminal Josephson junction devices with conductance measurements

The energy of an Andreev bound state in a clean normal metal in contact with two superconductors disperses with the difference Δϕ in the superconducting phase between the superconductors in much the same way as the energies of electrons in a one-dimensional crystal disperse with the crystal momentum k of the electrons. A normal metal with n superconductors maps onto a n − 1 dimensional crystal, each dimension corresponding to the phase difference Δϕi between a specific pair of superconductors. The resulting band structure as a function of the phase differences {Δϕi} in such ballistic devices has been proposed to have a topological nature with gapped regions characterized by different Chern numbers separated by regions where the gap in the quasiparticle spectrum closes. A similar complex evolution of the quasiparticle spectrum with {Δϕi} has also been predicted for diffusive normal metals in contact with multiple superconductors. While the underlying topological description is different in diffusive devices, gapped regions of the band diagram associated with different topological indices are also separated by regions where the gap closes. Here, we show that the variation of the density of states at the Fermi energy of such a system can be directly probed by relatively simple conductance measurements, allowing rapid characterization of the energy spectrum.

Theory of proximity effect in s+p -wave superconductor junctions

We derive a boundary condition for the Nambu Keldysh Green's function in diffusive normal metal / unconventional superconductor junctions applicable for mixed parity pairing. Applying this theory to a 1d model of $s+p$-wave superconductor, we calculate LDOS in DN and charge conductance of DN / $s+p$-wave superconductor junctions. When the $s$-wave component of the pair potential is dominant, LDOS has a gap like structure at zero energy and the dominant pairing in DN is even-frequency spin-singlet $s$-wave. On the other hand, when the $p$-wave component is dominant, the resulting LDOS has a zero energy peak and the dominant pairing in DN is odd-frequency spin-triplet $s$-wave. We show the robustness of the quantization of the conductance when the magnitude of $p$-wave component of the pair potential is larger than that of $s$-wave one. These results show the robustness of the anomalous proximity effect specific to spin-triplet superconductor junctions.

Generation of odd-frequency surface superconductivity with spontaneous spin current due to the zero-energy Andreev bound state

We propose that the odd-frequency $s$ wave ($s^{\rm{odd}}$ wave) superconducting gap function, which is usually unstable in the bulk, naturally emerges at the edge of $d$ wave superconductors. This prediction is based on the surface spin fluctuation pairing mechanism owing to the zero-energy surface Andreev bound state. The interference between bulk and edge gap functions triggers the $d+s^{\rm{odd}}$ state, and the generated spin current is a useful signal uncovering the ``hidden'' odd-frequency gap. In addition, the edge $s^{\rm{odd}}$ gap can be determined via the proximity effect on the diffusive normal metal. Furthermore, this study provides a decisive validation of the ``Hermite odd-frequency gap function,'' which has been an open fundamental challenge to this field.

Theory of the proximity effect in two-dimensional unconventional superconductors with Rashba spin-orbit interaction

We study the anomalous proximity effect in diffusive normal metal (DN)/unconventional superconductor junctions, where the local density of states (LDOS) in the DN has a zero-energy peak due to the penetration of the odd-frequency spin-triplet $s$-wave pairing. In this study, we consider a two-dimensional unconventional superconductor on the substrate in the presence of a Rashba spin-orbit coupling (RSOC) $\lambda$, where the Rashba vector is parallel to the $z$-direction. The anomalous proximity effect, originally predicted in spin-triplet $p$-wave superconductor junctions, is sensitive to the RSOC. It disappears with the increase of $\lambda$. On the other hand, the anomalous proximity effect can be switched on by the large $\lambda$ values in the spin-singlet $d_{xy}$-wave superconductor junctions. The resulting zero-energy LDOS and the magnitude of the odd-frequency spin-triplet $s$-wave pair amplitude increase with the increase of $\lambda$.

Coherent transport properties of a three-terminal hybrid superconducting interferometer

We present an exhaustive theoretical analysis of a double-loop Josephson proximity interferometer, as the one recently realized by Strambini et al. for the control of the Andreev spectrum via an external magnetic field. This system, called $\omega$-SQUIPT, consists of a T-shaped diffusive normal metal (N) attached to three superconductors (S) forming a double loop configuration. By using the quasiclassical Green function formalism, we calculate the local normalized density of states, the Josephson currents through the device and the dependence of the former on the length of the junction arms, the applied magnetic field and the S/N interface transparencies. We show that by tuning the fluxes through the double loop, the system undergoes transitions from a gapped to a gapless state. We also evaluate the Josephson currents flowing in the different arms as a function of magnetic fluxes and explore the quasi-particle transport, by considering a metallic probe tunnel-coupled to the Josephson junction and calculating its I-V characteristics. Finally, we study the performances of the $\omega$-SQUIPT and its potential applications, by investigating its electrical and magnetometric properties.

Decoherence of Cooper pairs and subgap magnetoconductance of superconducting hybrids

We demonstrate that electron–electron interactions fundamentally restrict the penetration length of superconducting correlations into a diffusive normal metal (N) attached to a superconductor (S). We evaluate the subgap magnetoconductance G of SN hybrids in the presence of electron–electron interactions and demonstrate that the effect of the magnetic field on G is twofold: it includes (i) additional temperature-independent dephasing of Cooper pairs and (ii) Zeeman splitting between the states with opposite spins. The dephasing length of Cooper pairs can be directly extracted from measurements of the subgap magnetoconductance in SN systems at low temperatures.

Spin supercurrent and phase-tunable triplet Cooper pairs via magnetic insulators

We demonstrate theoretically that a dissipationless spin current can flow a long distance through a diffusive normal metal by using superconductors interfaced with magnetic insulators. The magnitude of this spin supercurrent is controlled via the magnetization orientation of the magnetic insulators. The spin supercurrent obtained in this way is conserved in the normal metal just like the charge current and interestingly has a term that is independent of the superconducting phase difference. The quantum state of the system can be switched between 0 and $\ensuremath{\pi}$ by reversing the insulators from a parallel to antiparallel configuration with an external field. We show that the spin current is carried through the normal metal by superconducting triplet (odd-frequency) correlations and that the superconducting phase difference can be used to enhance these, leaving clear spectroscopic fingerprints in the density of states.

Dephasing of Cooper pairs and subgap electron transport in superconducting hybrids

We argue that electron-electron interactions fundamentally restrict the penetration length of Cooper pairs into a diffusive normal metal (N) from a superconductor (S). At low temperatures this Cooper pair dephasing length $L_\varphi$ remains finite and does not diverge at $T \to 0$. We evaluate the subgap conductance of NS hybrids in the presence of electron-electron interactions and demonstrate that this new length $L_\varphi$ can be directly extracted from conductance measurements in such structures.

Electron cooling in diffusive normal metal–superconductor tunnel junctions with a spin-valve ferromagnetic interlayer

We investigate heat and charge transport through a diffusive SIF${}_{1}$F${}_{2}$N tunnel junction, where N (S) is a normal (superconducting) electrode, I is an insulator layer, and F${}_{1,2}$ are two ferromagnets with arbitrary direction of magnetization. The flow of an electric current in such structures at subgap bias is accompanied by a heat transfer from the normal metal into the superconductor, which enables refrigeration of electrons in the normal metal. We demonstrate that the refrigeration efficiency depends on the strength of the ferromagnetic exchange field $h$ and the angle $\ensuremath{\alpha}$ between the magnetizations of the two F layers. As expected, for values of $h$ much larger than the superconducting order parameter $\ensuremath{\Delta}$, the proximity effect is suppressed and the efficiency of refrigeration increases with respect to a NIS junction. However, for $h\ensuremath{\sim}\ensuremath{\Delta}$ the cooling power (i.e., the heat flow out of the normal metal reservoir) has a nonmonotonic behavior as a function of $h$ showing a minimum at $h\ensuremath{\approx}\ensuremath{\Delta}$. We also determine the dependence of the cooling power on the lengths of the ferromagnetic layers, the bias voltage, the temperature, the transmission of the tunneling barrier, and the magnetization misalignment angle $\ensuremath{\alpha}$.

Inducing Odd-Frequency Triplet Superconducting Correlations in a Normal Metal

This work discusses theoretically the interplay between the superconducting and ferromagnetic proximity effects, in a diffusive normal metal strip in contact with a superconductor and a nonuniformly magnetized ferromagnetic insulator. The quasiparticle density of states of the normal metal shows clear qualitative signatures of triplet correlations with spin one (TCS1). When one goes away from the superconducting contact, TCS1 focus at zero energy under the form of a peak surrounded by dips, which show a typical spatial scaling behavior. This effect can coexist with a focusing of singlet correlations and triplet correlations with spin zero at finite but subgap energies. The simultaneous observation of both effects would enable an unambiguous characterization of TCS1.

Electron cooling by diffusive normal metal–superconductor tunnel junctions

We investigate heat and charge transport in ${\text{NN}}^{\ensuremath{'}}\text{IS}$ tunnel junctions in the diffusive limit. Here, N and S are massive normal and superconducting electrodes (reservoirs), ${\text{N}}^{\ensuremath{'}}$ is a normal metal strip, and I is an insulator. The flow of electric current in such structures at subgap bias is accompanied by heat transfer from the normal metal into the superconductor, which enables refrigeration of electrons in the normal metal. We show that the two-particle current due to Andreev reflection generates Joule heating, which is deposited in the N electrode and dominates over the single-particle cooling at low enough temperatures. This results in the existence of a limiting temperature for refrigeration. We consider different geometries of the contact, one dimensional and planar, which are commonly used in the experiments. We also discuss the applicability of our results to a double-barrier SINIS microcooler.

Odd-frequency pairing in superconducting heterostructures

We review the theory of odd-frequency pairing in superconducting heterostructures, wherean odd-frequency pairing component is induced near interfaces. A general description ofthe superconducting proximity effect in a normal metal or a ferromagnet attached to anunconventional superconductor (S) is given within quasiclassical kinetic theoryfor various types of symmetry state in S. Various possible symmetry classes ina superconductor are considered which are consistent with the Pauli principle:even-frequency spin-singlet even-parity (ESE) state, even-frequency spin-triplet odd-parity(ETO) state, odd-frequency spin-triplet even-parity (OTE) state and odd-frequencyspin-singlet odd-parity (OSO) state. As an example, we consider a junction betweena diffusive normal metal (DN) and a p-wave superconductor (even-frequencyspin-triplet odd-parity symmetry), where the pairing amplitude in DN belongs toan odd-frequency spin-triplet even-parity symmetry class. We also discuss themanifestation of odd-frequency pairing in conventional superconductor/normal (S/N)proximity systems and its relation to the classical McMillan–Rowell oscillations.

Phase sensitive electron-phonon coupling in a superconducting proximity structure

We study the role of the superconducting proximity effect on the electron-phonon energy exchange in diffusive normal metals (N) attached to superconductors (S). The proximity effect modifies the spectral response of the normal metal, in particular the local density of states. This leads to a weakening of the electron-phonon energy relaxation. We show that the effect is easily observable with modern thermometry methods, and predict that it can be tuned in structures connected to multiple superconductors by adjusting the phase difference between superconducting order parameters at the two NS interfaces.

Midgap Andreev resonant state affected by superconducting proximity effect of high-Tc cuprate attached to diffusive normal metal

We report how a midgap Andreev resonant state (MARS) is affected by the superconducting proximity effect at an interface of diffusive normal metal/insulator/dx.2_y2-wave superconductor (DN/I/DS) junctions. A zero-bias conductance peak (ZBCP) was observed for the (110)-oriented interface in Ag/SiO/Bi2Sr2CaCu2O8+δ (Bi-2212) planar junctions. The experimental ZBCP was analyzed by the extended circuit theory, which was improved by the introduction of spatial variations of the pair potential in a DS electrode near the junction interface. Our experimental results are well explained by the extended circuit theory for DN/I/DS junctions.

Dependence of Josephson Current on Orientational Angle in Diffusive p-Wave Superconductor Junctions

We calculate Josephson current in p-wave superconductor (P)/diffusive normal metal (DN)/insulator/diffusive normal metal/p-wave superconductor junctions, using quasi-classical Green’s function theory. We study the dependence of the critical current on the orientational angle at various temperatures. We find that the critical current in these junctions are proportional to cos αcos β. Here, α and β are angles between the direction perpendicular to the interfaces, located between the DN and P, and the crystal axis of left and right P respectively.

Theory of Josephson effect in Sr2RuO4/diffusive normal metal/Sr2RuO4 junctions

We derive a generalized Nazarov’s boundary condition for diffusive normal metal (DN)/chiral p-wave superconductor (CP) interface including the macroscopic phase of the superconductor. The Josephson effect is studied in CP/DN/CP junctions solving the Usadel equations under the above boundary condition. We find that, enhancement of a critical current at low temperature is small compared with that in px-wave /DN/px-wave junctions. As a result, temperature dependence of the critical current in these junctions is similar to that in conventional junctions. The result is consistent with the experiment in Sr2RuO4–Sr3RuO7 eutectic junctions. Similar feature is also found in current–phase relation.

Quasiclassical Green’s function theory of the Josephson effect in chiralp-wave superconductor/diffusive normal metal/chiralp-wave superconductor junctions

We study the Josephson effect between chiral p-wave superconductor / diffusive normal metal (DN) / chiral p-wave superconductor (CP/DN/CP) junctions using quasiclassical Green's function formalism with proper boundary conditions. The px+ipy-wave symmetry of superconducting order parameter is chosen which is believed to be a pairing state in Sr2RuO4. It is shown that the Cooper pairs induced in DN have an odd-frequency spin-triplet s-wave symmetry, where pair amplitude is an odd function of Matsubara frequency. Despite the peculiar symmetry properties of the Cooper pairs, the behavior of the Josephson current is rather conventional. We have found that the current phase relation is almost sinusoidal and the Josephson current is proportional to exp(-L/xi), where xi is the coherence length of the Cooper pair in DN and L is the length of DN. The Josephson current between CP / diffusive ferromagnet metal (DF) / CP junctions is also calculated. It is shown that the 0-pi transition can be realized by varying temperature or junction length L similar to the case of conventional s-wave junctions. These results may serve as a guide to study superconducting state of Sr2RuO4.

Theory of the Proximity Effect in Junctions with Unconventional Superconductors

We present a general theory of the proximity effect in junctions between diffusive normal metals (DN) and superconductors. Various possible symmetry classes in a superconductor are considered: even-frequency spin-singlet even-parity (ESE) state, even-frequency spin-triplet odd-parity state, odd-frequency spin-triplet even-parity (OTE) state and odd-frequency spin-singlet odd-parity state. It is shown that the pair amplitude in a DN belongs, respectively, to an ESE, OTE, OTE, and ESE pairing state since only the even-parity s-wave pairing is possible due to the impurity scattering.