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Physical review. B, Condensed matter

Order parameter of superfluid B3 near surfaces

Weiyi Zhang, J. Kurkijärvi, E. V. Thuneberg

Published Aug 1, 1987• 74 citations
Publisher

Abstract

The order parameter of superfluid $^{3}\mathit{B}$ is calculated near surfaces with the quasiclassical theory and a thin-dirty-layer surface-scattering model. The surface structures depend on the temperature, the surface roughness, the Fermi-liquid parameters, and the mass superflow tangential to the surface. All these are studied with special emphasis on the roughness and the superflow. A numerical technique of solving the quasiclassical equations is discussed.

DOI
10.1103/physrevb.36.1987

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Physical Review•Dec 13, 1965•284 citations
The order parameter of superfluid 3B^{3}\mathit{B}3B is calculated near surfaces with the quasiclassical theory and a thin-dirty-layer surface-scattering model. The surface structures depend on the temperature, the surface roughness, the Fermi-liquid parameters, and the mass superflow tangential to the surface. All these are studied with special emphasis on the roughness and the superflow. A numerical technique of solving the quasiclassical equations is discussed.
The order parameter and exchange field of superfluid 3B^{3}\mathit{B}3B near magnetically active surfaces with perfect reflection are calculated using the quasiclassical theory and a magnetically scattering surface model. Our results show that an exchange field is usually induced in the vicinity of such surfaces due to the spin rotation of the impending quasiparticles, but for certain magnetic configuration of the surface, the order parameters are not depressed at all. This is in strong contrast to the situation near nonmagnetic surfaces where the perpendicular order parameter always vanishes due to the triplet pairing nature of the Cooper pair. We discuss the possible application of this property to identify the Cooper pair symmetry of other unconventional superconductors.
This paper gives the boundary conditions for the quasiclassical theory of static superfluid Fermi systems in contact with an arbitrary reflecting surface. We demonstrate the application of the final closed circle of equations by displaying a self-consistent numerical solution of 3B^{3}\mathit{B}3B in contact with a ``randomly rough'' wall. The results exhibit how great the sensitivity of p-wave pairing is to surface scattering.
We discuss the hydrodynamic boundary condition for a superfluid moving tangentially to a rough surface. Specifically, we argue that the scattering of quantum fluctuations off surface roughness affects the nature of the boundary condition, and that this has important consequences including a theorized critical speed and the presence of normal fluid at any nonzero speed, even if the boundary is held at zero temperature (i.e., a moving superfluid flow creates a sustained temperature difference between the superfluid and the boundary). This hydrodynamic boundary condition is relevant not only for superfluid helium experiments but also for experiments with trapped dilute Bose--Einstein condensates, in particular, those involving atomic waveguides near surfaces.
We report the first numerical solution of the quasiclassical equations for superfluid 3B^{3}\mathit{B}3B flowing past a single specular wall. The superfluid gap function has (for finite flow) three nondegenerate components: The two components perpendicular to the flow are nearly unaffected by the flow; the component parallel to the flow is suppressed. We find that the critical current density given by our finite-temperature calculation is almost equal to the Ginzburg-Landau value down to temperatures of approximately T/Tc{T}_{c}Tc​=0.5.
The effects of the dipole interaction on the equilibrium and nonequilibrium properties of 3B^{3}\mathit{B}3B are calculated using the quasiclassical theory. In zero field, the dipole interaction distorts the l=1 gap, generates an l=3 gap, and induces a new rotated planar phase next to the normal-state transition. The field dependence of the Leggett angle ${\ensuremath{\theta}}_{L}$ is calculated to order (\ensuremath{\gamma}H/\ensuremath{\Delta})2{)}^{2})2, including the effect of nonquasiparticle corrections. Using the equilibrium properties as input, we calculate the zero-field NMR frequencies in the collisionless regime. A calculation of the field dependence of the longitudinal NMR frequency reveals the existence of a Fermi-liquid oscillation term due to the out-of-phase motion of the normal and superfluid components. Finally, we compare the collisionless and hydrodynamic theories of NMR.
The microscopic theory of a superfluid Fermi liquid at finite temperature is developed for the case of a pure system with SSS-wave pairing, and applied to the calculation of the static properties. As a function of $\ensuremath{\theta}\ensuremath{\equiv}\frac{T}{{T}_{c}}$ these properties are determined entirely by the Landau parameters F0{F}_{0}F0​, F1{F}_{1}F1​, Z0{Z}_{0}Z0​, etc., characterizing quasiparticle interactions in the normal phase. In particular the spin susceptibility $\ensuremath{\chi}$ and the density of the normal component ${\ensuremath{\rho}}_{n}$ are given by $\frac{\ensuremath{\chi}(\ensuremath{\theta})}{\ensuremath{\chi}(1)}=(1+\frac{1}{4}{Z}_{0})\frac{f(\ensuremath{\theta})}{[1+\frac{1}{4}{Z}_{0}f(\ensuremath{\theta})]},$ $\frac{{\ensuremath{\rho}}_{n}}{\ensuremath{\rho}}=(1+\frac{1}{3}{F}_{1})\frac{f(\ensuremath{\theta})}{[1+\frac{1}{3}{F}_{1}f(\ensuremath{\theta})]},$ where the universal function $f(\ensuremath{\theta})\ensuremath{\equiv}\ensuremath{-}{[\ensuremath{\nu}(0)]}^{\ensuremath{-}1}{\ensuremath{\Sigma}}_{\mathrm{p}}(\frac{\mathrm{dn}}{d{E}_{\mathrm{p}}})$ is the "effective density of states near the Fermi surface" relative to its value $\ensuremath{\nu}(0)$ in the normal phase. Thus the often-quoted expression ${\ensuremath{\rho}}_{n}=\frac{1}{3}{\ensuremath{\Sigma}}_{\mathrm{p}}{\mathrm{p}}^{2}(\frac{\mathrm{dn}}{d{E}_{\mathrm{p}}})$ is valid for an interacting system only in the limit $T\ensuremath{\rightarrow}0$. In the latter part of the paper a simple phenomenological theory of "Fermi-liquid" effects on $\ensuremath{\chi}$ and ${\ensuremath{\rho}}_{n}$ is developed for arbitrary conditions (including the presence of impurities and pairing with $l\ensuremath{\ne}0$); it is found that under most circumstances explicit expressions for $\ensuremath{\chi}$ and ${\ensuremath{\rho}}_{n}$ may be obtained which involve only the Landau parameters and a suitably generalized effective density of states. The theory should apply to the possible superfluid phase of He3{\mathrm{He}}^{3}He3 and to most superconductors. It is suggested that the Knight shift in nontransition-metal superconductors should display some "Fermi-liquid" effects. The weak-field dc penetration depth $\ensuremath{\lambda}(T)$ is shown to be insensitive to such effects both in the Pippard limit and near Tc{T}_{c}Tc​; however, in a London superconductor at lower temperatures the correction to $\ensuremath{\lambda}(T)$ should be observable and yield a direct estimate of F1{F}_{1}F1​.