Quasiclassical Formalism Research Articles

Extended Lawrence-Doniach model: The temperature evolution of the in-plane magnetic field anisotropy

Using the quasi-classical formalism we provide the description of the temperature and field-direction dependence of the in-plane upper critical field in layered superconductors, taking into account the interlayer Josephson coupling and the paramagnetic spin splitting. We generalize the Lawrence-Doniach model for the case of high magnetic fields and show that the re-entrant superconductivity is naturally described by our formalism when neglecting the Pauli pair breaking effect. We demonstrate that in layered superconductors the in-plane anisotropy of the onset of superconductivity exhibits four different\ temperature regimes: from the Ginzburg-Landau type in the vicinity of the critical temperature $T_{c0}$ with anisotropies of coherence lengths, up to the FFLO type induced by the strong interference between the modulation vector and the orbital effect. Our results are in agreement with the experimental measurements of the field-angle dependence of the superconducting onset temperature of the organic compound (TMTSF)$_{2} $ClO$_{4}$.

Andreev reflection at half-metal/superconductor interfaces with nonuniform magnetization

Andreev reflection at the interface between a half-metallic ferromagnet and a spin-singlet superconductor is possible only if it is accompanied by a spin flip. Here we calculate the Andreev reflection amplitudes for the case that the spin flip originates from a spatially non-uniform magnetization direction in the half metal. We calculate both the microscopic Andreev reflection amplitude for a single reflection event and an effective Andreev reflection amplitude describing the effect of multiple Andreev reflections in a ballistic thin film geometry. It is shown that the angle and energy dependence of the Andreev reflection amplitude strongly depends on the orientation of the gradient of the magnetization with respect to the interface. Establishing a connection between the scattering approach employed here and earlier work that employs the quasiclassical formalism, we connect the symmetry properties of the Andreev reflection amplitudes to the symmetry properties of the anomalous Green function in the half metal.

First-principles-baseds±-wave modeling for iron-based superconductors: Specific heat and nuclear magnetic relaxation rate

In order to consistently explain controversial experimental results on superconducting states observed by different probes in typical iron-based superconductors, we construct a realistic multiband ${s}^{\ifmmode\pm\else\textpm\fi{}}$-wave pairing model by combining the quasiclassical formalism with the first-principles calculation. The model successfully resolves the controversies in contrast to the fact that simplified models such as two-band ${s}^{\ifmmode\pm\else\textpm\fi{}}$-wave one fail to do. A key in the model is the existence of relatively small gaps which leads to material-dependent peculiarities.

Crossed Andreev reflection at spin-active interfaces

With the aid of the quasiclassical Eilenberger formalism we develop a theory of non-local electron transport across three-terminal ballistic normal-superconducting-normal (NSN) devices with spin-active NS interfaces. The phenomenon of crossed Andreev reflection (CAR) is known to play the key role in such transport. We demonstrate that CAR is highly sensitive to electron spins and yields a rich variety of properties of non-local conductance which we describe non-perturbatively at arbitrary voltages, temperature, spin-dependent interface transmissions and their polarizations. Our results can be applied to multi-terminal hybrid structures with normal, ferromagnetic and half-metallic electrodes and can be directly tested in future experiments.

Phase States of Multiterminal Mesoscopic Normal-Metal–Superconductor Structures

We study a mesoscopic normal-metal structure with four superconducting contacts, two of which are joined into a loop. The structure undergoes transitions between three (meta)stable states, with different phase configurations triggered by nonequilibrium conditions. These transitions result in spectacular changes in the magnetoresistance. We find a qualitative agreement between the experiments and a theory based on the quasiclassical Keldysh formalism.

Crossed Andreev reflection in diffusive contacts: Quasiclassical Keldysh-Usadel formalism

Crossed Andreev reflection in multiterminal structures in the diffusive regime is addressed within the quasiclassical Keldysh-Usadel formalism. The elastic cotunneling and crossed Andreev reflection of quasiparticles give nonlocal currents and voltages (depending on the actual biasing of the devices) by virtue of the induced proximity effect in the normal-metal electrodes. The magnitude of the nonlocal processes is found to scale with the square of the barrier transparency and to decay exponentially with interface spacing. Nonlocal cotunneling and crossed Andreev conductances are found to contribute equally to the nonlocal current, which is of relevance to the use of normal-metal--superconducting heterostructures as sources of entanglement.

Theory of charge transport in diffusive normal metal/unconventional superconductor junctions

Novel theory of charge transport which is available for both a diffusive normal metal/unconventional singlet superconductor (DN/USS) and a diffusive normal metal/triplet superconductor (DN/TS) junctions is presented based on the Keldysh–Nambu quasiclassical Green's function formalism. For DN/USS junctions, proximity effect in DN region and the mid gap Andreev resonant state (MARS) formed at the USS side of the DN/USS interface strongly compete each other. On the other hand, for DN/TS junctions, proximity effect is enhanced by the MARS. In DN/TS junction, the total resistance is drastically reduced and is completely independent of the resistance of the DN in the extreme case. We can also check this dramatic effect by numerical simulations without using quasiclassical formalism.

Mixed-State Thermodynamics of Superconductors with Moderately Large Paramagnetic Effects

Effects of Pauli paramagnetism on thermodynamic quantities in a vortex state, such as the specific heat $C$ and magnetization $M$, are studied using the quasiclassical Eilenberger formalism. We demonstrate that with an increase of paramagnetic depairing effect, the sigh of the curvature of the field dependence of $C$ changes from negative to positive, and that the Maki parameter $\kappa_2$ becomes an increasing function of temperature. Our results provide a natural explanation for the unusual field dependence of $C$ seen in CeCoIn$_5$ in terms of the paramagnetic effect.

Quasiclassical Theory for Thermodynamic Properties of the Two-Band Superconductor MgB2 under Magnetic Fields

Consistent account is provided of strongly anisotropic properties in thermodynamics of MgB$_2$ under magnetic field $H$ using approximate analytic solution in the quasiclassical formalism. With the strength of the pairing interactions and the Fermi velocities of two bands as fitting parameters, it is possible to reproduce in wide regions of the phase diagram the $H$- and temperature dependences of relevant quantities such as the zero-energy density of states (ZEDOS), two energy gaps, the specific heat and the $H_{c2}$ anisotropy. It is demonstrated that the velocity ratio of two bands affects considerably the low-field behavior of the ZEDOS and energy gaps. The ratio also affects the high-temperature behavior of the $H_{c2}$ anisotropy. The anisotropy of the coherence length in the $\sigma$ band depends weakly on $H$ and $T$, whereas the coherence length in the $\pi$ band remains isotropic even in the presence of the inter-band coupling.

Anisotropy of In-Plane Magnetization Due to Nodal Gap Structure in Vortex State

We examine the interplay between anisotropy of the in-plane magnetization and the nodal gap structure on the basis of the approximate analytic solution in the quasiclassical formalism. We show that a four-fold oscillation appears in the magnetization, and its amplitude changes sign at an intermediate field. The high-field oscillation originates from the anisotropy of the upper critical field, while the low-field behavior can be understood by the thermally activated quasiparticles near nodes depending on the applied field angles. The temperature dependence of the magnetization also shows a similar sign change. The anisotropy of the magnetization offers a possible measurement to identify the gap structure directly for a wide class of type II superconductors.

Anisotropic Diamagnetic Response in Type-II Superconductors with Gap and Fermi-Surface Anisotropies

The effects of anisotropic gap structures on a diamagnetic response are investigated in order to demonstrate that the field-angle-resolved magnetization [ML(chi)] measurement can be used as a spectroscopic method to detect gap structures. Our microscopic calculation based on the quasiclassical Eilenberger formalism reveals that ML(chi) in a superconductor with a fourfold gap displays a fourfold oscillation reflecting the gap and Fermi-surface anisotropies, and the sign of this oscillation changes at a field between Hc1 and Hc2. As a prototype of unconventional superconductors, magnetization data for borocarbides are also discussed.

Influence of Gap Structures to Specific Heat in Oriented Magnetic Fields: Application to the Orbital Dependent Superconductor, Sr2RuO4

We discuss influence of modulation of gap function and anisotropy of Fermi velocity to field angle dependences of upper critical field, $H_{c2}$, and specific heat, $C$, on the basis of the approximate analytic solution in the quasiclassical formalism. Using 4-fold modulation of the gap function and the Fermi velocity in the single-band model, we demonstrate field and temperature dependence of oscillatory amplitude of $H_{c2}$ and $C$. We apply the method to the effective two-band model to discuss the gap structure of Sr$_2$RuO$_4$, focusing on recent field angle-resolved experiments. It is shown that the gap structures with the intermediate magnitude of minima in $[100]$ direction for $\gamma$ band, and tiny minima of gaps in $[110]$ directions for $\alpha$ and $\beta$ bands give consistent behaviors with experiments. The interplay of the above two gaps also explains the anomalous temperature dependence of in-plane $H_{c2}$ anisotropy, where the opposite contribution from the passive $\alpha\beta$ band is pronounced near $T_c$.

Quasiparticle decoherence in d-wave superconducting qubits.

It is usually argued that the presence of gapless quasiparticle excitations at the nodes of the d-wave superconducting gap should strongly decohere the quantum states of a d-wave qubit, making quantum effects practically unobservable. Using a self-consistent linear response nonequilibrium quasiclassical formalism, we show that this is not necessarily true. We find quasiparticle conductance of a d-wave grain boundary junction to be strongly phase dependent. Midgap states as well as nodal quasiparticles contribute to the conductance and therefore decoherence. Quantum behavior is estimated to be detectable in a qubit containing a d-wave junction with appropriate parameters.

Microscopic nonequilibrium theory of double-barrier Josephson junctions

We study nonequilibrium charge transport in a double-barrier Josephson junction, including nonstationary phenomena, using the time-dependent quasiclassical Keldysh Green's function formalism. We supplement the kinetic equations by appropriate time-dependent boundary conditions and solve the time-dependent problem in a number of regimes. From the solutions, current-voltage characteristics are derived. It is understood why the quasiparticle current can show excess current as well as deficit current and how the subgap conductance behaves as function of junction parameters. A time-dependent nonequilibrium contribution to the distribution function is found to cause a nonzero averaged supercurrent even in the presence of an applied voltage. Energy relaxation due to inelastic scattering in the interlayer has a prominent role in determining the transport properties of double-barrier junctions. Actual inelastic scattering parameters are derived from experiments. It is shown as an application of the microscopic model, how the nature of the intrinsic shunt in double-barrier junctions can be explained in terms of energy relaxation and the opening of Andreev channels.

Nonequilibrium quasiclassical theory for Josephson structures

We present a nonequilibrium quasiclassical formalism suitable for studying linear-response ac properties of Josephson junctions. The nonequilibrium self-consistency equations are satisfied, to a very good accuracy, already in zeroth iteration. We use the formalism to study ac Josephson effect in a ballistic superconducting point contact. The real and imaginary parts of the ac linear conductance are calculated both analytically (at low frequencies) and numerically (at arbitrary frequency). They show strong temperature, frequency, and phase dependences. Many anomalous properties appear near $\ensuremath{\varphi}=\ensuremath{\pi}.$ We ascribe them to the presence of zero energy bound states.

Polarization rotation in parametric scattering of polaritons in semiconductor microcavities

Polarization of light emitted by a semiconductor microcavity in the regime of a resonant parametric scattering of the exciton polaritons shows extremely strong and unusual dependence on the polarization of pumping light. This dependence is interpreted here using the pseudospin model and in the framework of a quasiclassical formalism where the parametric scattering is described as resonant four-wave mixing. We show that the optically induced splitting of the exciton-polariton eigenstate, both in linear and circular polarizations, is responsible for the observed polarization effects. The splitting in circular polarizations, achieving 0.5 meV, has been detected experimentally, while the splitting in linear polarizations, which is much weaker, only manifests itself in the pseudospin dynamics of the exciton polaritons.

Energy spectrum of a confined two-dimensional particle in an external magnetic field

The energy spectrum of a particle confined within a circle in the presence of an external magnetic field perpendicular to the plane of confinement is studied both exactly and approximately by the quasiclassical formalism (Wentzel–Kramers–Brillouin). For pure spatial confinement (without magnetic field) the energy spectrum for states other than the ground state is twofold degenerate, while in the case of pure magnetic confinement the spectrum shows the infinite-fold degeneration, typical of Landau states. For both types of confinement, the latter infinite-fold degeneration is lifted due to spatial confinement. Interestingly enough, for a given ratio between spatial and magnetic confining lengths, the magnetic flux is quantized. The conditions for the quantization of the flux are established; the nature and peculiarity of the energy spectrum are also discussed.

Field theory of mesoscopic fluctuations in superconductor-normal-metal systems

Thermodynamic and transport properties of mesoscopic conductors are strongly influenced by the proximity of a superconductor: an interplay between the large scale quantum coherent wave functions in the normal mesoscopic and the superconducting region, respectively, leads to unusual mechanisms of quantum interference. These manifest themselves in both the mean and the mesoscopic fluctuation behaviour of superconductor-normal-metal (SN) hybrid systems being strikingly different from those of conventional mesoscopic systems. After reviewing some established theories of SN quantum interference phenomena, we introduce a new approach to the analysis of SN mesoscopic physics. Essentially, our formalism represents a unification of the quasi-classical formalism for describing mean properties of SN systems on the one hand, with more recent field theories of mesoscopic fluctuations on the other hand. Thus, by its very construction, the new approach is capable of exploring both averaged and fluctuation properties of ...

Algebraically Self-Consistent Quasiclassical Approximation on Phase Space

The Wigner–Weyl mapping of quantum operators to classical phase space functions preserves the algebra, when operator multiplication is mapped to the binary “*” operation. However, this isomorphism is destroyed under the quasiclassical substitution of * with conventional multiplication; consequently, an approximate mapping is required if algebraic relations are to be preserved. Such a mapping is uniquely determined by the fundamental relations of quantum mechanics, as is shown in this paper. The resultant quasiclassical approximation leads to an algebraic derivation of Thomas–Fermi theory, and a new quantization rule which—unlike semiclassical quantization—is non-invariant under action transformations of the Hamiltonian, in the same qualitative manner as the true eigenvalues. The quasiclassical eigenvalues are shown to be significantly more accurate than the corresponding semiclassical values, for a variety of 1D and 2D systems. In addition, certain standard refinements of semiclassical theory are shown to be easily incorporated into the quasiclassical formalism.

Quasiclassical formalism and the Andreev equation

In this article I shall make explicit the connection between the quasiclassical Green’s function and the Andreev quasiparticle energies and wavefunctions. The physical meaning of the components of the Green’s function is elucidated.