Magnetic irreversibility and relaxation in assembly of ferromagnetic nanoparticles

Measurements of the magnetic irreversibility line and time-logarithmic decay of the magnetization are described for three $Fe_{2}O_{3}$ samples composed of regular amorphous, acicular amorphous and crystalline nanoparticles. The relaxation rate is the largest and the irreversibility temperature is the lowest for the regular amorphous nanoparticles. The crystalline material exhibits the lowest relaxation rate and the largest irreversibility temperature. We develop a phenomenological model to explain the details of the experimental results. The main new aspect of the model is the dependence of the barrier for magnetic relaxation on the instantaneous magnetization and therefore on time. The time dependent barrier yields a natural explanation to the time-logarithmic decay of the magnetization. Interactions between particles as well as shape and crystalline magnetic anisotropies define a new energy scale that controls the magnetic irreversibility. Introducing this energy scale yields a self-consistent explanation of the experimental data.Comment: RevTex, 16 JPEG figures, to appear in PR

Orbital upper critical field and its anisotropy of clean one- and two-band superconductors

The Helfand-Werthamer (HW) scheme\cite{HW} of evaluating the orbital upper critical field is generalized to anisotropic superconductors in general, and to two-band clean materials, in particular. Our formal procedure differs from those in the literature; it reproduces not only the isotropic HW limit, but also the results of calculations for the two-band superconducting MgB$_2$\cite{MMK,DS} along with the existing data on $H_{c2}(T)$ and its anisotropy $\gamma(T)=H_{c2,ab}(T)/H_{c2,c}(T)$ ($a,c$ are the principal directions of a uniaxial crystal). Using rotational ellipsoids as model Fermi surfaces we apply the formalism developed to study $\gamma(T)$ for a few different anisotropies of the Fermi surface and of the order parameters. We find that even for a single band d-wave order parameter $\gamma(T)$ decreases on warming, however, relatively weakly. For order parameters of the form $\Delta(k_z) = \Delta_0(1+\eta\cos k_za)$,\cite{Xu} according to our simulations $\gamma(T)$ may either increase or decrease on warming even for a single band depending on the sign of $\eta$. Hence, the common belief that the multi-band Fermi surface is responsible for the temperature variation of $\gamma$ is proven incorrect

Interband coupling and transport interband scattering in s±s_{\pm} superconductors

A two-band model with repulsive interband coupling and interband {\it transport} (potential) scattering is considered to elucidate their effects on material properties. In agreement with previous work, we find that the bands order parameters $\Delta_{1,2}$ differ and the large is at the band with a smaller normal density of states (DOS), $N_{n2}<N_{n1}$. However, the bands energy gaps, as determined by the energy dependence of the DOS, are equal due to scattering. For each temperature, the gaps turn zero at a certain critical interband scattering rate, i.e. for strong enough scattering the model material becomes gappless. In the gapless state, the DOS at the band 2 is close to the normal state value, whereas at the band 1 it has a V-shape with non-zero minimum. When the normal bands DOS' are mismatched, $N_{n1}\ne N_{n2}$, the critical temperature $T_c$ is suppressed even in the absence of interband scattering, $T_c(N_{n1})$ has a dome-like shape. With increasing interband scattering, the London penetration depth at low temperatures evolves from being exponentially flat to the power-law and even to near linear behavior in the gapless state, the latter being easily misinterpreted as caused by order parameter nodes

Effective Demagnetization Factors of Diamagnetic Samples of Various Shapes

Effective demagnetizing factors that connect the sample magnetic moment with the applied magnetic field are calculated numerically for perfectly diamagnetic samples of various non-ellipsoidal shapes. The procedure is based on calculating total magnetic moment by integrating the magnetic induction obtained from a full three dimensional solution of the Maxwell equations using adaptive mesh. The results are relevant for superconductors (and conductors in AC fields) when the London penetration depth (or the skin depth) is much smaller than the sample size. Simple but reasonably accurate approximate formulas are given for practical shapes including rectangular cuboids, finite cylinders in axial and transverse field as well as infinite rectangular and elliptical cross-section strips.Comment: updated version, corrected typos et

Orbital upper critical field of type-II superconductors with pair breaking

The orbital upper critical field $H_{c2}$ is evaluated for isotropic materials with arbitrary transport and pair-breaking scattering rates. It is shown that unlike transport scattering which enhances $H_{c2}$, the pair breaking suppresses the upper critical field and reduces the dimensionless ratio $h^*(0)=H_{c2}(0)/T_c(dH_{c2}/dT)_{T_c}$ from the Helfand-Werthamer value of $\approx 0.7$ to 0.5 for a strong pair-breaking. $h^*(T)$ is evaluated for arbitrary transport and pair-breaking scattering. A phenomenological model for the pair-breaking suppression by magnetic fields is introduced. It shows qualitative features such as a positive curvature of $H_{c2}(T)$ and the low temperature upturn usually associated with multi-band superconductivity

Effect of equatorial line nodes on upper critical field and London penetration depth

The upper critical field $H_{c2}$ and its anisotropy are calculated for order parameters with line nodes at equators, $k_z=0$, of the Fermi surface of uniaxial superconductors. It is shown that characteristic features found in Fe-based materials -- a nearly linear $H_{c2}(T)$ in a broad $T$ domain, a low and increasing on warming anisotropy $\gamma_H= H_{c2,ab}/ H_{c2,c}\,$ -- can be caused by competing effects of the equatorial nodes and of the Fermi surface anisotropy. For certain material parameters, $\gamma_H(T)-1$ may change sign on warming in agreement with recorded behavior of FeTeS system. It is also shown that the anisotropy of the penetration depth $\gamma_\lambda= \lambda_c/\lambda_{ab}$ decreases on warming to reach $\gamma_H$ at $T_c$ in agreement with data available. For some materials $\gamma_\lambda(T)$ may change on warming from $\gamma_\lambda>1$ at low $T$s to $\gamma_\lambda<1$ at high $T$s

Anisotropic criteria for the type of superconductivity

The classical criterion for classification of superconductors as type-I or type-II based on the isotropic Ginzburg-Landau theory is generalized to arbitrary temperatures for materials with anisotropic Fermi surfaces and order parameters. We argue that the relevant quantity for this classification is the ratio of the upper and thermodynamic critical fields, $H_{c2}/H_c$, rather than the traditional ratio of the penetration depth and the coherence length, $\lambda/\xi$. Even in the isotropic case, $H_{c2}/H_c$ coincides with $\sqrt{2}\lambda/\xi$ only at the critical temperature $T_c$ and they differ as $T$ decreases, the long known fact. Anisotropies of Fermi surfaces and order parameters may amplify this difference and render false the criterion based on the value of $\kappa=\lambda/\xi$

Collective flux creep: beyond the logarithmic solution

Numerical studies of the flux creep in superconductors show that the distribution of the magnetic field at any stage of the creep process can be well described by the condition of spatial constancy of the activation energy $U$ independently on the particular dependence of $U$ on the field B and current $j$. This results from a self-organization of the creep process in the undercritical state $j<j_{c}$ related to a strong non-linearity of the flux motion. Using the spatial constancy of $U$, one can find the field profiles $B(x)$, formulate a semi-analytical approach to the creep problem and generalize the logarithmic solution for flux creep, obtained for $U=U(j)$, to the case of essential dependence of $U$ on $B$. This approach is useful for the analysis of dynamic formation of an anomalous magnetization curve (''fishtail''). We analyze the quality of the logarithmic and generalized logarithmic approximations and show that the latter predicts a maximum in the creep rate at short times, which has been observed experimentally. The vortex annihilation lines (or the sample edge for the case of remanent state relaxation), where B=0, cause instabilities (flux-flow regions) and modify or even destroy the self-organization of flux creep in the whole sample.Comment: 12 pages, 9 PS figure

Response to Comment by A. Bussmann-Holder (arXiv:0909.3603)

Response to Comment by A. Bussmann-Holder (arXiv:0909.3603

London Penetration Depth and Pair Breaking

The London penetration depth is evaluated for isotropic materials for any transport and pair-breaking Born scattering rates. Besides known results, a number of new features are found. The slope $|d\rho/d\theta |$ of the normalized superfluid density $\rho=\lambda^2(0)/\lambda^2(\theta)$ at the transition $\theta=T/T_c=1$ has a minimum near the value of the pair-breaking parameter separating gapped and gapless states. The low-$T$ exponentially flat part of $\rho$ for the s-wave materials is suppressed by increasing pair breaking. For strong $T_c$ suppression by magnetic impurities the "Homes scaling" $\lambda^{-2}(0) \propto \sigma T_c$ with $\sigma$ being the normal conductivity gives way to $\lambda^{-2}(0) \propto \sigma T_c^2$. For the d-wave order parameter, the transport and spin-flip Born scattering rates enter the theory only as a sum, in particular, they affect the $T_c$ depression in the same manner. We confirm that the linear low temperature behavior of $\rho$ in a broad range of the combined scattering parameter turns to the $T^2$ behavior only when the critical temperature is suppressed at least by a factor of 3 relative to the clean limit $T_{c0}$. Moreover, in this range, $\rho (\theta)$ is only weakly dependent on the scattering parameter, i.e. it is nearly universal