The London penetration depth $lambda$ is the basic length scale for electromagnetic behavior in a superconductor. Precise measurements of $lambda$ as a function of temperature, field, and impurity scattering have been instrumental in revealing the na
ture of the order parameter and pairing interactions in a variety of superconductors discovered over the past decades. Here we recount our development of the tunnel-diode resonator technique to measure $lambda$ as a function of temperature and field in small single crystal samples. We discuss the principles and applications of this technique to study unconventional superconductivity in the copper oxides and other materials such as iron-based superconductors. The technique has now been employed by several groups worldwide as a precision measurement tool for the exploration of new superconductors.
Temperature-dependent London penetration depth, $lambda(T)$, of a high quality optimally-doped $text{YBa}_{2}text{Cu}_{3}text{O}_{7-delta}$ single crystal was measured using tunnel-diode-resonator technique. Controlled artificial disorder was induced
by low-temperature (20~K) irradiation by 2.5 MeV electrons at two large doses of $3.8times10^{19}$and $5.3times10^{19}$ electrons per $textrm{cm}^{2}$. The irradiation caused significant suppression of the superconductors critical temperature, $T_{c}$, from 94.6 K to 90.0 K, and to 78.7 K, respectively. The low-temperature behavior of $lambdaleft(Tright)$ evolves from a $T-$ linear in pristine state to a $T^{2}-$ behavior after irradiation, expected for a line-nodal $d-$wave superconductor. However, the original theory that explained such behavior assumed a unitary limit of the scattering potential, whereas usually in normal metals and semiconductors, Born scattering is sufficient to describe the experiment. To estimate the scattering potential strength, we calculated the superfluid density, $rho_{s}=lambda^{2}left(0right)/lambda^{2}left(Tright)$, varying the amount and strength of non-magnetic scattering using a self-consistent $t-$matrix theory. Comparing experimental and theoretical coefficients $A$ and $B$ of the low-temperature power series, $rho_{s}approx1-At-Bt^{2}$, we determine the amplitude of the scattering phase shift to be around 65$^{o}$. Knowing this value is important for further theoretical analysis of the microscopic mechanisms of superconductivity in $text{YBa}_{2}text{Cu}_{3}text{O}_{7-delta}$ high$-T_{c}$ superconductor.
Closed-topology magnetic domains are usually observed in thin films and in an applied magnetic field. Here we report the observation of rectangular cross-section tubular ferromagnetic domains in thick single crystals of CeAgSb2 in zero applied field.
Relatively low exchange energy, small net magnetic moment, and anisotropic in-plane crystal electric fields lower the domain wall energy and allow for the formation of the closed-topology patterns. Upon cycling the magnetic field, the domain structure irreversibly transforms into a dendritic open-topology pattern. This transition between closed and open topologies results in a topological magnetic hysteresis - the actual hysteresis in magnetization, not due to the imperfections and pinning, but due to the difference in the pattern morphology. Similar physics was suggested before in pure type-I superconductors and is believed to be a generic feature of other nonlinear multi-phase systems in the clean limit.
Radio-frequency (14.6 MHz) AC magnetic susceptibility, $chi^{prime}_{AC}$, of dytio was measured using a self-oscillating tunnel-diode resonator. Measurements were made with the excitation AC field parallel to the superimposed DC magnetic field up 5
T in a wide temperature range from 50 mK to 100 K. At 14.6 MHz a known broad peak of $chi^{prime}_{AC}(T)$ from kHz - range audio-frequency measurements around 15~K for both [111] and [110] directions shifts to 45~K, continuing the Arrhenius activated behavior with the same activation energy barrier of $E_a approx 230$~K. Magnetic field dependence of $chi^{prime}_{AC}$ along [111] reproduces previously reported low-temperature two-in-two-out to three-in-one-out spin configuration transition at about 1~T, and an intermediate phase between 1 and 1.5~T. The boundaries of the intermediate phase show reasonable overlap with the literature data and connect at a critical endpoint of the first-order transition line, suggesting that these low-temperature features are frequency independent. An unusual upturn of magnetic susceptibility at $T to 0$ was observed in magnetic fields between 1.5~T and 2~T for both magnetic field directions, before fully polarized configuration sets in above 2~T.
Static (DC) and dynamic (AC, at 14 MHz and 8 GHz) magnetic susceptibilities of single crystals of a ferromagnetic superconductor, $textrm{EuFe}_{2}(textrm{As}_{1-x}textrm{P}_{x})_{2}$ (x = 0.23), were measured in pristine state and after different do
ses of 2.5 MeV electron or 3.5 MeV proton irradiation. The superconducting transition temperature, $T_{c}(H)$, shows an extraordinarily large decrease. It starts at $T_{c}(H=0)approx24:textrm{K}$ in the pristine sample for both AC and DC measurements, but moves to almost half of that value after moderate irradiation dose. Our results suggest that in $textrm{EuFe}_{2}(textrm{As}_{1-x}textrm{P}_{x})_{2}$ superconductivity is affected by local-moment ferromagnetism mostly via the spontaneous internal magnetic fields induced by the FM subsystem. Another mechanism is revealed upon irradiation where magnetic defects created in ordered $text{Eu}^{2+}$ lattice act as efficient pairbreakers leading to a significant $T_{c}$ reduction upon irradiation compared to other 122 compounds. On the other hand, the exchange interactions seem to be weakly screened by the superconducting phase leading to a modest increase of $T_{m}$ (less than 1 K) after the irradiation drives $T_{c}$ to below $T_{m}$. The results suggest that FM and SC phases coexist microscopically in the same volume.
Magnetic susceptibility of non-ellipsoidal samples is a long-standing problem in experimental studies of magnetism and superconductivity. Here the quantitative description of the Meissner-London response (no Abrikosov vortices) of right circular cyli
nders in an axial magnetic field is given. The three-dimensional adaptive finite-element modeling was used to calculate the total magnetic moment, m, in a wide range of London penetration depth, lambda, to sample size ratios. By fitting the numerical data, the closed-form universal magnetic susceptibility is formulated involving only sample dimensions and lambda, thus providing a recipe for determining the London penetration depth from the accurate magnetic susceptibility measurements. Detailed examples of the experimental data analysis using the developed approach are given. The results can be extended to the frequently used cuboid-shaped samples.
A simple procedure to extract anisotropic London penetration depth components from the magnetic susceptibility measurements in realistic samples of cuboidal shape is described.
We study the effects of anisotropic order parameters on the temperature dependence of London penetration depth anisotropy $gamma_lambda(T)$. After MgB$_2$, this dependence is commonly attributed to distinct gaps on multi-band Fermi surfaces in superc
onductors. We have found, however, that the anisotropy parameter may depend on temperature also in one-band materials with anisotropic order parameters $Delta(T,k_F)$, a few such examples are given. We have found also that for different order parameters, the temperature dependence of $Delta(T)/Delta(0)$ can be represented with good accuracy by the interpolation suggested by D. Einzel, J. Low Temp. Phys, {bf 131}, 1 (2003), which simplifies considerably the evaluation of $gamma_lambda(T)$. Of particular interest is mixed order parameters of two symmetries for which $gamma_lambda(T)$ may go through a maximum for a certain relative weight of two phases. Also, for this case, we find that the ratio $Delta_{max}(0)/T_c$ may exceed substantially the weak coupling limit of 1.76. It, however, does not imply a strong coupling, rather it is due to significantly anisotropic angular variation of $Delta$.
In-plane anisotropy of electrical resistivity was studied in samples of the hole-doped Ba$_{1-x}$K$_x$Fe$_2$As$_2$ in the composition range $0.21 leq x leq 0.26$ where anisotropy changes sign. Low-temperature ($sim$20~K) irradiation with relativistic
2.5 MeV electrons was used to control the level of disorder and residual resistivity of the samples. Modification of the stress-detwinning technique enabled measurements of the same samples before and after irradiation, leading to conclusion of anisotropic character of predominantly inelastic scattering processes. Our main finding is that the resistivity anisotropy is of the same sign irrespective of residual resistivity, and remains the same in the orthorhombic $C_2$ phase above the re-entrant tetragonal transition. Unusual $T$-linear dependence of the anisotropy $Delta rho equiv rho_a(T)-rho_b(T)$ is found in pristine samples with $x=$0.213 and $x=$0.219, without similar signatures in either $rho_a(T)$ or $rho_b(T)$. We show that this feature can be reproduced by a phenomenological model of R.~M.~Fernandes {it et al.} Phys. Rev. Lett. {bf 107},217002 (2011). We speculate that onset of fluctuations of nematic order on approaching the instability towards the re-entrant tetragonal phase contributes to this unusual dependence.
The anisotropic London equations taking into account the normal currents are derived and applied to the problem of the surface impedance in the Meisner state of anisotropic materials. It is shown that the complex susceptibility of anisotropic slab de
pends on the orientation of the applied microwave field relative to the crystal axes. In particular, the anisotropic sample in the microwave field is subject to a torque, unless the field is directed along with one of the crystal principle axes.
